Optimal octave stretching/compressing

I’m guessing someone has done this already. I’m interested in alternate tunings, but not so well connected to the community.

Since stretched octaves are well tolerated, or even preferred, and since some non-octave tunings have advantages over similar EDO tunings, is there, for a given equal temperament, an optimal way to stretch or compress the octave to minimize overall dissonance?

You could imagine in 15-EDO, for example, some compromise between 1200 cent octaves (15-EDO with bad fifths) and 1170 cent octaves (Wendy Carlos alpha scale with bad octaves). This is an extreme example, but probably any equal temperament could be improved by stretching or compressing the octaves a little.

I’ve already worked on this a little bit, but some of the finer points get a little complicated. There’s no reason for me to go any further down that road if it’s already been perfected by someone else.

Should I keep working on it, or is it considered a solved problem already?

Hi Scott,

The question of optimal octave stretch has been studied here.
We have methods to compute optimal stretch for any regular
temperament... not just EDOs, but also higher-rank temperaments
represented by MOS scales (aka well-formed scales) and so on.
The two most popular of these methods are called TOP tuning
and TE tuning. They are very similar and give similar results.

TOP is lucidly explained by Paul Erlich, who discovered it
http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

TE is implemented in Graham Breed's online temperament finder
http://x31eq.com/temper/
This tool will actually tell you the optimal stretch for
every temperament known to man (and some that are yet unknown!).

-Carl

At 10:22 AM 3/15/2012, you wrote:
>
>I�m guessing someone has done this already. I�m interested in
>alternate tunings, but not so well connected to the community.
>
>Since stretched octaves are well tolerated, or even preferred, and
>since some non-octave tunings have advantages over similar EDO
>tunings, is there, for a given equal temperament, an optimal way to
>stretch or compress the octave to minimize overall dissonance?
>
>You could imagine in 15-EDO, for example, some compromise between 1200
>cent octaves (15-EDO with bad fifths) and 1170 cent octaves (Wendy
>Carlos alpha scale with bad octaves). This is an extreme example, but
>probably any equal temperament could be improved by stretching or
>compressing the octaves a little.
>
>I�ve already worked on this a little bit, but some of the finer points
>get a little complicated. There�s no reason for me to go any further
>down that road if it�s already been perfected by someone else.
>
>Should I keep working on it, or is it considered a solved problem
>already?
>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Hi Scott,
>
> The question of optimal octave stretch has been studied here.
> We have methods to compute optimal stretch for any regular
> temperament... not just EDOs, but also higher-rank temperaments
> represented by MOS scales (aka well-formed scales) and so on.
> The two most popular of these methods are called TOP tuning
> and TE tuning. They are very similar and give similar results.
>
> TOP is lucidly explained by Paul Erlich, who discovered it
> http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

--I don't get it. This reads vaguely like old mystical manuscripts on the "music of the spheres," astrology, complicated diagrams showing how to read minds from the shape of lines on your palm, and how the solar system is made of icosahedrons.

Instead, it seems to me, such claims need to be based on psychoacoustic research, which I'm not seeing any of in this paper.

Is there, in fact, any evidence that octaves ought to be stretched?

On Thu, Mar 15, 2012 at 3:02 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> --I don't get it. This reads vaguely like old mystical manuscripts on the
> "music of the spheres," astrology, complicated diagrams showing how to read
> minds from the shape of lines on your palm, and how the solar system is made
> of icosahedrons.
>
> Instead, it seems to me, such claims need to be based on psychoacoustic
> research, which I'm not seeing any of in this paper.

You can see a short writeup of Paul's work on his harmonic entropy
model of consonance, which is based on the notion that time resolution
leads to uncertainty in the frequency domain (and hence that all dyads
are uncertain), here

http://sethares.engr.wisc.edu/paperspdf/HarmonicEntropy.pdf

The idea is that, as a result, the incoming signal can be better
represented as a superposition of ratios than as a single ratio, and
that the more any one ratio predominates, the more consonant the sound
is. I'm not entirely sure that that's all there is to it, but it's a
good start and definitely a strong effort to root some notable aspect
of music theory in psychoacoustics.

I've generalized Paul's model to use Renyi Entropy instead of Shannon
entropy and implemented it online - you can find an app here:

www.mikebattagliamusic.com/HE-JS/HE.html

> Is there, in fact, any evidence that octaves ought to be stretched?

What sort of evidence are you looking for, and what do you mean by
"ought to be?" There are lots of people here who like the sound of
stretched octaves (although I don't care too much about them, myself).

-Mike

Warren wrote:

>> TOP is lucidly explained by Paul Erlich, who discovered it
>> http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf
>
>--I don't get it. This reads vaguely like old mystical manuscripts on
>the "music of the spheres," astrology, complicated diagrams showing
>how to read minds from the shape of lines on your palm, and how the
>solar system is made of icosahedrons.

One of the things you should have learned before writing this

http://dl.dropbox.com/u/3507527/MusicTh.html

is how to distinguish Paul's papers from these mystical manuscripts.

>Is there, in fact, any evidence that octaves ought to be stretched?

There can never be any evidence that any interval "ought to be
stretched". It is a fact that in temperament, some intervals
must be stretched. Paul shows that, rather than excluding octaves
from a temperament optimization, they can be included using a
weighting scheme that is consistent with psychoacoustic evidence.
For those who would rather exclude them entirely, there is
POTE tuning

http://xenharmonic.wikispaces.com/Tenney-Euclidean+tuning#Pure%20octaves%20TE%20tuning

-Carl

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

> For those who would rather exclude them entirely, there is
> POTE tuning
>
> http://xenharmonic.wikispaces.com/Tenney-Euclidean+tuning#Pure%20octaves%20TE%20tuning

And for those who want to stretch, there are a variety of approaches, some of which are:

http://xenharmonic.wikispaces.com/TOP+tuning

http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning#TE tuning

http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning#The Frobenius projection map

http://xenharmonic.wikispaces.com/The+Riemann+Zeta+Function+and+Tuning#The Z function

well (to reply to MB & CL), I suppose one could argue if some intervals
must be stretched (which they must) then it makes sense to claim that octaves are just another interval and hence there is no reason they should have special unstretchable status; the optimum could be to distort it also trying for the optimum (i.e. smallest in some
overall psychoacoustic sense) combination of distortions for all intervals.

>> Is there, in fact, any evidence that octaves ought to be stretched?

>What sort of evidence are you looking for, and what do you mean by
>"ought to be?" There are lots of people here who like the sound of
>stretched octaves (although I don't care too much about them, myself).

--well, the "evidence" would have to be exactly that "people like the sound."
But not just anecdotally -- I mean real evidence that more people really do like
the sound. Or if not "like" then some other quantitatively solidly measured advantage of some sort, e.g. if they listen to stretched octave music for a while then they perform better on some acoustic test shortly afterward than a control group, statistically significantly.

Re astrology and palmistry, the Ehlich paper I was pointed to, seemed to me to have entirely too much theoretical and utterly baseless fantasizing -- tons of tables and diagrams all constructed purely from the power of pure thought, just as the fantasies of solar system arising from octahedra, complete with detailed artist illustrations were "constructed" -- and with little, maybe zero, basis in actual experimental evidence cited/used in said paper, that I noticed (although I suppose in the case of the solar system fantasies there actually was a small amount of genuine numerical evidence, in unwelcome contrast). The ratio of the two struck me as astonishingly large.

As a sanity check to try --
suppose Ehlich had written a basically similar paper but in a different field such as, say, weather prediction. Well, I suggest to you, that in most other areas, a paper like that would have been laughed out of the journal. If things are that way, it is not a good sign.
I haven't tried to digest it in very careful detail so what I'm saying could be
too much criticism too early, but, for what it is worth, that was my impression.

On Thu, Mar 15, 2012 at 3:48 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> well (to reply to MB & CL), I suppose one could argue if some intervals
> must be stretched (which they must) then it makes sense to claim that
> octaves are just another interval and hence there is no reason they should
> have special unstretchable status; the optimum could be to distort it also
> trying for the optimum (i.e. smallest in some
> overall psychoacoustic sense) combination of distortions for all
> intervals.

Right. On the other hand, one could argue that the octave does have
special significance, because there's some sense in which pitches that
are separated by an octave are equivalent to one another. But, on the
other hand, there are people who have spent years in the Bohlen-Pierce
scale (13 equal divisions of 3/1 with a "diatonic" scale of 2 1 1 2 1
2 1 2 1) who claim that in that context, they hear 3/1 as fulfilling
that function too.

The literature on these sorts of topics isn't terribly conclusive
because there's not much literature on it.

> >> Is there, in fact, any evidence that octaves ought to be stretched?
>
> >What sort of evidence are you looking for, and what do you mean by
> >"ought to be?" There are lots of people here who like the sound of
> >stretched octaves (although I don't care too much about them, myself).
>
> --well, the "evidence" would have to be exactly that "people like the
> sound."
> But not just anecdotally -- I mean real evidence that more people really
> do like
> the sound. Or if not "like" then some other quantitatively solidly
> measured advantage of some sort, e.g. if they listen to stretched octave
> music for a while then they perform better on some acoustic test shortly
> afterward than a control group, statistically significantly.

It doesn't matter much to me if people like the sound at first. What's
important to me is if people like the sound after they've spent some
time training themselves to get used to it. And that's obviously
something you can't show unless you do a proper longitudinal study
with good controls - preferably a lot of longitudinal studies. This
sort of research has obviously never happened since the field is so
new, but that's the hand we're dealt. If the puzzle were already
solved, we wouldn't be here.

> Re astrology and palmistry, the Ehlich paper I was pointed to, seemed to
> me to have entirely too much theoretical and utterly baseless fantasizing --
> tons of tables and diagrams all constructed purely from the power of pure
> thought, just as the fantasies of solar system arising from octahedra,
> complete with detailed artist illustrations were "constructed" -- and with
> little, maybe zero, basis in actual experimental evidence cited/used in said
> paper, that I noticed (although I suppose in the case of the solar system
> fantasies there actually was a small amount of genuine numerical evidence,
> in unwelcome contrast). The ratio of the two struck me as astonishingly
> large.

Can you give a concrete example? How much of the paper did you read?

-Mike

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:

> > TOP is lucidly explained by Paul Erlich, who discovered it
> > http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf
>
> --I don't get it. This reads vaguely like old mystical manuscripts on the "music of the spheres," astrology, complicated diagrams showing how to read minds from the shape of lines on your palm, and how the solar system is made of icosahedrons.

This claim strikes me as somewhat bizarre. Can you point to something specific which strikes you as analogous to astrology or palmistry? Unattached to anything specific, it's hard to see where you are coming from here.

Actually, long as I started to rant, let me do a bit more in a more general manner.

It seems to me, your overriding goal is to figure out the best musical tuning system.
(Right?)

Now toward that end, you've built all kinds of highly sophisticated mathematical theories, connecting to Riemann zeta function, number theory, lattices, and using all kinds
of weird coined terminology, and on and on -- albeit, far as I can tell, not reaching any clear conclusion to the Quest. (By "you" I mean, the set of people behind this yahoogroup,
mostly.)

Now my reaction to this, which is somewhat just an opinion from an outsider, is this. You're overtheorizing and underexperimenting.

I suspect you don't need all that fancy theory. Let's say we throw it all in the trash.
I instead suspect that if you knew (i.e. had a subroutine) saying exactly what "best" meant, then you wouldn't need the theory, in fact it it'd just get in your way; you instead could probably find the best tuning system just by brute force computation (well, somewhat intelligently written brute force...) in way less time than it took to cook all that other stuff up. Game over.

If so, then there is only ONE question that really matters: what is the quality metric
by which we measure what is the "best" tuning system?

If we knew that, then I conjecture the rest would be brute force computation.

And this One Question is simply not answerable by the power of pure thought.
It is a psycho-acousto-biological question. It needs to
be answered by doing quantitative experiments using real people and real
equipment. And to some extent this plus modeling the results has already been done, e.g
the "harmonic entropy" measure MB was recently mentioning. Quite possibly
the truth is somewhat uglier and less elegant --reality-based data usually is -- but
nothing stops you from making a quality measure based entirely on measured data
with very little theorizing or modeling needed there either.

These experiments are the crux. In their absence, all the rest is garbage.

So anyhow, this, it seems to me, is the straight line path toward your Goal, but you
do not seem to be taking that path.

Here's a simple sanity check. Has there ever been an experiment where people were asked to rate music in equi-tempered scale versus some (allegedly superior) slightly-unequal
tempered system, with clear statistically significant results saying A is better than B?
I haven't been looking at this recently, and I'm somewhat ignorant...
but back when I did, I do not recall finding any such clear study. There were some studies that seemed to me to go partway in that direction.

And if we haven't got that, not even for a single tuning system, anywhere, ever, then
why the hell are you wasting your time with all this ultra-theoretical sophisticated garbage? You haven't yet reached square one. It's like trying to develop subatomic physics
before you've even figured out how to rub two sticks together. Before being ready for
that, you have to reach, like, square ten, i.e. being able to detect & quantitatively measure clear quality differences between, say, 10 different reasonable tuning systems. This may or may not be feasible to detect. If it isn't, again I ask why the heck you are wasting your time. If it is, I ask, why the heck this experiment is not being done as task #1.

Well... that seems enough of a rant to stimulate considerable flaming. I'll
put on my asbestos vest now.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@> wrote:
>
> > > TOP is lucidly explained by Paul Erlich, who discovered it
> > > http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf
> >
> > --I don't get it. This reads vaguely like old mystical manuscripts on the "music of the spheres," astrology, complicated diagrams showing how to read minds from the shape of lines on your palm, and how the solar system is made of icosahedrons.
>
> This claim strikes me as somewhat bizarre. Can you point to something specific which strikes you as analogous to astrology or palmistry? Unattached to anything specific, it's hard to see where you are coming from here.

--well good grief, the entire paper seemed to be absolutely screaming it. Its kind
of like jumping into an a ocean, complaining you are wet, then somebody says "cant you
give me some specific example of wetness?"

And on an unrelated(?) note, I point out that at another time,
Lerdahl & Jackendoff wrote a 1983 tome "generative theory of tonal music"
which was much fawned over. OK it has now been almost 30 years. Has the L&J theory
been followed up on, accomplished anything, or what? I suggest to you, that from my perspective (and I think any reasonable person's perspective, by which I mean, anybody who is not in bed with L&J) it has been a total failure. It sounded really sophisticated on the surface, but there wasn't any experimental backing/underpinning or for that matter any clarity in their thinking. It was just hot air.

Warren wrote:

>It seems to me, your overriding goal is to figure out the best musical
>tuning system.
>(Right?)

That's not what most people here are getting at, no. We found ways
of measuring the tradeoffs of different tuning systems.

>Now toward that end, you've built all kinds of highly sophisticated
>mathematical theories, connecting to Riemann zeta function, number
>theory, lattices, and using all kinds of weird coined terminology,

True, but it's all minutiae on top of a very simple model, called
the "regular mapping paradigm". And haven't you posted about some
pretty arcane stuff yourself?

>Now my reaction to this, which is somewhat just an opinion from an
>outsider, is this. You're overtheorizing and underexperimenting.

What experiments would you suggest? As an art, music enjoys
terrifying variety... "preferences" seem to be the most an
experiment can reveal.

>I suspect you don't need all that fancy theory.
>Let's say we throw it all in the trash.

I have a better idea: We start over and you ask where the resources
for beginners are.

>If we knew that, then I conjecture the rest would be brute force
>computation.

You're on a list where the results of many brute force computations
have been shared. I think it would be a good time for you to quit
while you're ahead and actually read Erlich's papers.

-Carl

On Thu, Mar 15, 2012 at 4:55 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> Now my reaction to this, which is somewhat just an opinion from an
> outsider, is this. You're overtheorizing and underexperimenting.

That's because you're not on the right group. Most of the
experimentation is happening on the Facebook "Xenharmonic Alliance"
group. Most of the people here are on there too, but those groups
serve to act as a testbed for much of the theory here.

If you had thrown this criticism out there a year or two ago, I'd have
totally agreed, but you're a year too late, I'm afraid. And it's no
mystery that, partly from what's been seen over there, that there's
still plenty of work to do over here. (This is why I messaged you
offlist, in fact.)

Anyway, we all tried to get you to join these groups before, but you
said you wanted to stay on tuning-math. This is like an offshoot of
the main tuning list, which itself has mostly moved to XA, so it was
odd that you initially wanted to stay here. Maybe now's a good time to
check the other groups out.

> If so, then there is only ONE question that really matters: what is the
> quality metric
> by which we measure what is the "best" tuning system?
>
> If we knew that, then I conjecture the rest would be brute force
> computation.

Keenan Pepper's done some work on finding scales that minimize HE. The
best one for harmony overall ended up being meantone, which has a
generator of half a meantone fourth. Unfortunately, I don't care much
about this because it means that Westerners ended up already finding
"the best scale." Well, what's "best" to me is finding something
-new-.

Other than that, if the goal is to have as many harmonic ratios in a
temperament as possible, and to tune them as accurately as possible,
Graham's implemented his temperament finder here:

http://x31eq.com/temper/pregular.html

Here's a search for the best tempered systems containing
approximations to all ratios within the multiplicative group of primes
up to 11:

http://x31eq.com/cgi-bin/pregular.cgi?limit=11&error=5.0

Each of these temperaments, like "Orwell," implies a range of scales
and characteristic chord progressions (which may not be apparent if
you don't know how to read the page). So this tool can be
extraordinarily useful for learning how to make music (go over to XA
to hear the music that people are making with this stuff, but also
stuff outside the theory).

Anyway, the problem is that these metrics don't adequately represent,
to me, all of the things I like about a tuning system or a scale. If
they did, I wouldn't still be here. There are other scales with crap
for harmony that have the most beautiful logical, organizational,
"conceptual," "functional," whatever-you-want layouts I've ever seen,
and it's a worthwhile tradeoff for me in those systems to accept some
beating in the harmony. So if what you want is more relevant things to
model, and more "finders" to find the best tunings that have those
things - then yes, I agree. So what do we model?

> And this One Question is simply not answerable by the power of pure
> thought.
> It is a psycho-acousto-biological question. It needs to
> be answered by doing quantitative experiments using real people and real
> equipment.

The informal listening experiments you suggest are what led us to
where we are now.

There were lots of informal listening experiments done for things like
harmonic entropy, which itself was modeled after the results of much
more formal psychoacoustics papers.

Informal listening experiments invariably lead to problems. In this
case, the problem is that you simply can't gain any quality
information about the susceptibility of perception to adaptive changes
with exposure from some touch-and-go listening tests. The result is
that after the "reboot" of this community over on XA, there are many
more people who are messing with high-error tunings, and things other
than ratios, than there were on this list.

Informal listening experiments done on the population also creating
the theory can invariably lead to a sort of theoretical inbreeding.
Notably, it fails to account for the possibility that, if any basic
process exists where a listener can adapt to find music that he or she
is exposed to to be more palatable over time, theorists will become
habitually exposed and adapt to the products of their own theories,
which will circularly reinforce the validity of those theories in
predicting their perception - since part of the prediction now
includes their own expectations. So if you want to account for this
phenomenon, you need to do it on a lot of different people and across
different cultures.

> Here's a simple sanity check. Has there ever been an experiment where
> people were asked to rate music in equi-tempered scale versus some
> (allegedly superior) slightly-unequal
> tempered system, with clear statistically significant results saying A is
> better than B?
> I haven't been looking at this recently, and I'm somewhat ignorant...
> but back when I did, I do not recall finding any such clear study. There
> were some studies that seemed to me to go partway in that direction.

This is a poor design for a listening test. It will fail to measure
anything other than what sort of music Western listeners are most
susceptible to liking at the current point in time. It doesn't really
give you any insight into the nature of music cognition, because it
fails to track how Western listeners might adapt to novel tunings
after some time has elapsed. You'd need to do a proper longitudinal
study for that.

Listening tests that fit into the "come up with stuff and see if
people like it right away" view just suck. They obviously will fail to
account for tuning systems that produce an averse reaction to Western
ears at first, but which yield and become quite palatable once you
settle into them after a week or so.

Lastly, you simply can't write off a tuning system with this sort of
experimental design, because you can never be sure that the listening
examples that you're bombarding the listeners' ears with are
representative of the "best" sorts of compositions available in that
system. If some musical device works better in one tuning system than
others, you can't be sure that your chosen representative device isn't
just one element in a class of similar devices of which another would
work.

So proper experimental design is far from the trivial matter that you
seem to imply.

> And if we haven't got that, not even for a single tuning system, anywhere,
> ever, then
> why the hell are you wasting your time with all this ultra-theoretical
> sophisticated garbage?

Because we obviously don't have any money to do these tests, and we
need to move forward as best we can. Also, your experimental design is
a poor one to measure "the best tuning system ever." But if the gist
of this message is "there's more to music than ratios," then yes, I
agree. And I think quite a few of us do at this point, but we're not
sure where to go next.

Maybe this would be a good time to finally repost here all of the
stuff I posted about melody on XA now.

-Mike

On Thu, Mar 15, 2012 at 5:03 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...>
> wrote:
> >
> > This claim strikes me as somewhat bizarre. Can you point to something
> > specific which strikes you as analogous to astrology or palmistry?
> > Unattached to anything specific, it's hard to see where you are coming from
> > here.
>
> --well good grief, the entire paper seemed to be absolutely screaming it.
> Its kind
> of like jumping into an a ocean, complaining you are wet, then somebody
> says "cant you
> give me some specific example of wetness?"

So can you point out a concrete example of a statement made in that
paper which has no theoretical basis? Although I think there's some
new ideas these days which diverge from the paradigm presented in that
paper, I'd be hard pressed to say that the whole thing has no
theoretical basis. It just seems more like your level of comprehension
of the paper is rather low overall.

-Mike

Warren wrote:

>And on an unrelated(?) note, I point out that at another time,
>Lerdahl & Jackendoff wrote a 1983 tome "generative theory of tonal music"
>which was much fawned over. OK it has now been almost 30 years. Has
>the L&J theory been followed up on, accomplished anything, or what?
>I suggest to you, that from my perspective (and I think any reasonable
>person's perspective, by which I mean, anybody who is not in bed with L&J)
>it has been a total failure. It sounded really sophisticated on the
>surface, but there wasn't any experimental backing/underpinning or for
>that matter any clarity in their thinking. It was just hot air.

I've never heard of L&J so I can't comment. Most of the academic
music theory I'm aware of is no better than numerology, but I'll
look into it at my next opportunity.

To be a little more specific on your earlier question about the
best tuning system, we can actually answer this given a couple
assumptions and the values of a few parameters.

The assumptions are that the music to be tuned contains harmonies
(plenty of music doesn't) and that the tuning of these harmonies
matters at all. If so, one can choose a badness function. There
are two main choices: Cangwu badness, which needs an error threshold,
and logflat badness, which needs a complexity cutoff. In both cases
you'll need to say how error and complexity are measured. The error
metric choices are things like TOP, TE, and POTE that we were just
discussing. In practice they give similar results (TOP and TE are
the Linf and L2 versions of the same thing). You'll also need to
say what prime limit you want to consider.

That's enough to get an answer, and as I mentioned, plenty of these
scenarios have been brute-forced here (you'll find some results in
the Erilch paper you keep blabbing about). These choices represent
maybe 1 byte of freedom, but many of them will give the same answer
for the best tuning system.

Actually the answer given is an abstract regular temperament, which
will have an optimal tuning under the error metric you chose. But
you still have to pick a scale size, which is a whole number.

The main degrees of freedom here are the prime limit, scale size,
and value of the badness parameter. The reason I said most people
here are not on a Quest is because we've found that these **really
are degrees of freedom** (they amount to personal preference).
Interesting music has already been recorded in several systems
that would come out "best" under different choices.

None of this touches melody, which is another big area where tuning
is important.

If you really don't like Erlich, you might try my Too-Condensed
Tuning-Math Outline
http://lumma.org/music/theory/tctmo/

-Carl

I don't like the "professionals can be trained to appreciate it" standard, I prefer the "uninitiated ordinary people prefer it" standard.
Because with the former standard, virtually anything perceivable would qualify.

Also, I'm interested in what music reveals about the nature of the human mind.

Re some other stuff, I did read Erlich and you should stop complaining about that,
otherwise I'll say how the fact you haven't read Lehrdahl+Jackendoff, which was clearly the premier work in the field, proves you are an utter incompetent. (Actually I do not believe that, I'm just saying if that's the road you want to take, then that's where you'd be.)

Some of the responses posted here (e.g. by MB) to me seem good. I fully agree
with him what I said was not a great experimental design, nor was it intended to be, but I also say that if the naive experiment isn't doable, then you're clearly in trouble.

I also don't like experiments made on small groups of insiders, I agree with whoever was
saying that's pretty meaningless. I also agree doing good experiments takes money and effort. But it seems to be the crux, so until/unless done you're in trouble.

If you really can't see a single thing in Erlich's paper that might have stimulated my "strange" reaction then I assure you that me pointing one out, would not change that.

I would like it if you posted more info here about the experimrnts you said had been done, which I was not aware of.

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:

> If you really can't see a single thing in Erlich's paper that might have stimulated my "strange" reaction then I assure you that me pointing one out, would not change that.

Without a single specific example and critique this is just bullshit, and you are intelligent enough to produce something of actual substance instead. I'm asking, again, but I will call bullshit on bullshit.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@> wrote:
>
> > If you really can't see a single thing in Erlich's paper that might have stimulated my "strange" reaction then I assure you that me pointing one out, would not change that.
>
> Without a single specific example and critique this is just bullshit, and you are intelligent enough to produce something of actual substance instead. I'm asking, again, but I will call bullshit on bullshit.

--ok, since you absolutely insist, how was (every decimal of) the number "1089.38" justified in the TOP Wurschmidt horogram by experimental evidence?

How about the sentence "Concordance here is a POSITIVE phenomenon which reflects the extent to which combinations of notes can elicit the same gestalt phenomena that apply to the study of single notes"?

How about "the pythagorean comma 531441:524288"?
I assure you there is zero experimental support for every last decimal there, this is
pure fantasy that this has any significance.

[Note: I got these 3 examples by tossing a dart at the paper 8 times and picking three of the 8 impact sites. They have no significance aside from that.]

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:

> How about "the pythagorean comma 531441:524288"?
> I assure you there is zero experimental support for every last
> decimal there, this is pure fantasy that this has any significance.

Now you're just revealed as a complete tool. Are you going
to shape up or shall I just ban you and get it over with?

-Carl

On Thu, Mar 15, 2012 at 6:14 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> I don't like the "professionals can be trained to appreciate it" standard,
> I prefer the "uninitiated ordinary people prefer it" standard.
> Because with the former standard, virtually anything perceivable would
> qualify.

If it's generally true that people tend to develop some meaningful
sort of affinity to things that they're used to, then that would seem
to be a fundamental tenet of how music works (and something which
easily fits into your "debugging" thing), so I can't imagine why on
Earth you'd want to dismiss that sort of insight (if it's true) a
priori.

That could suggest, among other things, that part of the experience of
music derives from, in addition to any inherent pleasure the stimuli
themselves may bring, the "meaningful" manipulation of those stimuli
on a "symbolic" level. It could also suggest that it's not the stimuli
themselves that are important but things like how they fulfill and
violate the listeners' predictions in a pleasing way. David Huron is
known for advancing this sort of paradigm.

Anyways, the above idea might be true, or it might be false. But the
real point is that we can't just a priori assume that it's false.
We'll will forever remain completely ignorant to this possibility as
long as we devise experiments that dismiss it a priori, don't control
for factors relevant to these things, and then assume that these
experiments are enough to dictate "how music works" in a general
sense.

Also, the a priori dismissal of the relevance of training-related
factors are irrelevant would equate to the same sort of unfounded
music-cognitive assumption that you're accusing Paul for making. I was
actually hoping that your "debugging" paradigm would help us to
understand these sorts of training-related phenomena more, since I'm
now firmly convinced they're here to stay.

Finally, my personal experience is that this sort of affinity-building
is very real and that naive experiments such as the one you suggested
end up missing this completely and unnecessarily limit the realm of
"good" tunings.

> Some of the responses posted here (e.g. by MB) to me seem good. I fully
> agree
> with him what I said was not a great experimental design, nor was it
> intended to be, but I also say that if the naive experiment isn't doable,
> then you're clearly in trouble.

The naive experiment has led us to where we are now and I'm not 100%
satisfied with it.

Also, you need to recognize that if you're going to be performing
these sorts of experiments on Westerners, then your sample population
has already had a lifetime of exposure to Western music. This means
that your sample population itself consists of what you called
"trained professionals," so it's not like you aren't doing these
experiments on tabula rasae. (I certainly wouldn't call this a
population of "trained professionals" myself, but this is using the
standard that you yourself set above.)

There are known correlates to training in the psychoacoustics and
psychology literature. For an example of one, I suggest you check out
some literature on the "categorical perception" of musical intervals,
where musicians trained in a tuning system (like 12-EDO) tend to hear
the continuous spectrum of dyads as being quantized into different
versions of discrete interval "categories." It's the same sort of
principle that takes place with phonemes, and leads to one developing
an "accent" in a foreign language. This phenomenon generally increases
with musical training and I've anecdotally observed differences in the
ways that Western musicians and Western-nonmusicians react to
microtonal music that's well in line with the literature on this
subject (and with my own experience).

In short, microtonal music may sound "weird" to highly trained 12-EDO
musicians for the same reason that a foreign accent may sound "weird"
to highly-trained English speakers. And developing a perfect accent in
a foreign language is difficult (but not impossible). Gauging the ease
with which highly-trained musicians can retailor their conceptual
framework for music to a new tuning system is of prime interest in my
world right now.

Anyway, experiments like the one you proposed end up steamrolling over
all of that. They only reveal one thing, which is what microtonal
scales that Westerners with lifelong exposure to Western music will
prefer on first listen. This experiment will partially measure the
extent of Western conditioning on the perception of music, and as such
fail to satisfy your criteria for providing a platform on which to
build a psychoacoustic/biological foundation for music all by itself.

But, if you don't claim that your experiment reveals more than that,
then that's certainly a fair enough thing to try and get some data on.
It's just that that's not the primary focus of the group, at least not
for me. I'm more concerned with what sorts of novel musical -systems-
are possible, even if it means a bit more adaptation is required. (A
thorough understanding of the nature of this sort of adaptation would
unite our two aims by possibly creating a measure of the "cognitive
distance" from one tuning to another.)

However, your proposed experiment still fails even for that. As I said
before, if people hate some musical example you give in some tuning
and like it in some other one, you STILL can't write it off. All you
can write off is your own compositional ability in that tuning at this
point in time.

You still haven't proven that there isn't a way to "unlock" the sound
of the tuning by using it in a better way. My experience is that there
often is a "better way" than what you'd assume at first glance.

> Also, I'm interested in what music reveals about the nature of the human
> mind.

Good. Then I hope I've convinced you that cross-sectional studies to
test these sorts of things are utterly useless to generalize about the
nature of the human mind, and that longitudinal studies are what's
really needed.

Anyway, the above is a really long reply, but hopefully thorough. If
you don't want me to write such long replies in the future, please
don't write about how simple it is to devise music cognition
experiments in the future, or I'll go nuts :)

> Re some other stuff, I did read Erlich and you should stop complaining
> about that,
> otherwise I'll say how the fact you haven't read Lehrdahl+Jackendoff,
> which was clearly the premier work in the field, proves you are an utter
> incompetent. (Actually I do not believe that, I'm just saying if that's the
> road you want to take, then that's where you'd be.)

Didn't you just finish saying how irrelevant Lehrdahl+Jackendoff were,
and how nobody takes them seriously? Now they're the premier work in
the field? I've read a great deal of music cognition literature at
this point and I don't think I've seen them cited once. They've been
on my to-read list for a few months though.

> I also don't like experiments made on small groups of insiders, I agree
> with whoever was
> saying that's pretty meaningless. I also agree doing good experiments
> takes money and effort. But it seems to be the crux, so until/unless done
> you're in trouble.

In fact, we are in trouble. The solution is to make as -little-
assumptions as possible and do what we can with what we have. One
assumption that's decently solid is that when a stimulus is
near-periodic, it activates various psychoacoustic features that can
be harnessed for musical purposes. That's the one assumption that's
powered all of the theory that's come out of here. That's also the
only assumption, to my knowledge, contained in Paul's paper. I happen
to think there are more postulates we need to make to move forward,
and I'm having a hell of a time figuring out how to narrow those down.

> If you really can't see a single thing in Erlich's paper that might have
> stimulated my "strange" reaction then I assure you that me pointing one out,
> would not change that.

You are a mathematician. You ought to be able to give an example of a
specific claim that does not logically follow from the premises set
forth in the document. And if you claim that one of the premises
themselves are unfounded, you ought to be able to give a specific
example of one.

> I would like it if you posted more info here about the experimrnts you
> said had been done, which I was not aware of.

These are, again, completely informal. One thing I did was this:

http://soundcloud.com/mikebattagliamusic/sets/the-categorical-experiments/

This contains the same composition, retuned so that the size ratio of
the large to the small step varies from 1:1 (7-EDO) to Infinity
(5-EDO). I wanted to see if this piece remained intelligible
throughout, or if there'd be a sudden break in intelligibility when
the ratio of the major third changed from 5/4 to 9/7. Everyone was
able to recognize the melody all throughout the spectrum, except that
it started to break down and get unintelligible at the extremes (like
72-EDO and 5-EDO, and 75-EDO 7-EDO). From this I concluded that it's
important to consider how easy it is to tell the intervals in a scale
apart from one another. It's obviously easy to figure out if two
intervals are a half step apart if the half step is 100 cents, and it
gets harder as the half step get closer to 0 cents - I doubt most
people could tell two intervals apart if they're separated by a "half
step" of 1 cent, and there are clear psychoacoustic reasons for this
which should satisfy your psychoacoustic/biological criteria. I'm
currently working on a few different models of this "scale clarity"
which I hope to post here when I get a bit more time to finish up.

People who spent a lot of time with JI tended to report hearing 9/7 as
being "a different sort of thing" as 5/4, with tunings like 27-EDO
being like a variation of the original. People who didn't, but had
lots of 12-EDO musical training, tended to report hearing 9/7 and 5/4
as both "types of major thirds." There was at least one person who was
really into JI (Pete Kosmorsky) who said the experiment changed his
concept of what the "identity" of an interval is. There were also
people like Chris Vaisvil who didn't understand WTF I was testing or
why on earth it would be significant that a warped diatonic scale
sounds exactly like a warped diatonic scale.

Many people took this as a quiz to see which one they liked. I thought
17-EDO was best. A few people liked 22-EDO and 27-EDO best because it
was "exotic." Gene liked 31-EDO best, which was most harmonically
accurate. A lot of people hated the "flat" tunings, like the ones
closer to 7-EDO, but Keenan notably liked them a lot. Kalle liked
12-EDO best.

I'd like to do this again someday and do the test blind this time. I
think it'd be a useful data point, but I still wouldn't claim it's
enough to consider the matter settled, not by a long shot.

That's just one experiment I did, there are lots of others that have
been done here too.

-Mike

On Thu, Mar 15, 2012 at 7:11 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> --ok, since you absolutely insist, how was (every decimal of) the number
> "1089.38" justified in the TOP Wurschmidt horogram by experimental evidence?

The TOP tuning minimizes the maximum weighted error in the tuning.
Minimizing the max-weighted error is good because it leads to
temperaments that approximate ratios as accurately as possible. That's
the generator which minimizes it. I don't know what experimental
evidence you're looking for.

> How about the sentence "Concordance here is a POSITIVE phenomenon which
> reflects the extent to which combinations of notes can elicit the same
> gestalt phenomena that apply to the study of single notes"?

What's wrong with that?

> How about "the pythagorean comma 531441:524288"?
> I assure you there is zero experimental support for every last decimal
> there, this is
> pure fantasy that this has any significance.

What decimal? The Pythagorean comma is the ratio 531441/524288, which
is the difference between 12 justly tuned 3/2's and 7 justly tuned
2/1's, known since antiquity.

-Mike

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Warren wrote:
>
> >> TOP is lucidly explained by Paul Erlich, who discovered it
> >> http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf
> >
> >--I don't get it. This reads vaguely like old mystical manuscripts on
> >the "music of the spheres," astrology, complicated diagrams showing
> >how to read minds from the shape of lines on your palm, and how the
> >solar system is made of icosahedrons.
>
> One of the things you should have learned before writing this
>
> http://dl.dropbox.com/u/3507527/MusicTh.html
>
> is how to distinguish Paul's papers from these mystical manuscripts.

I don't see the necessity to discredit Warren's entire hypothesis as worthless simply because he doesn't understand the value in some of Paul's ideas. This is a very snarky remark on your behalf Carl, if it isn't already dull to say the least.

Ryan

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@> wrote:

> --ok, since you absolutely insist, how was (every decimal of) the number "1089.38" justified in the TOP Wurschmidt horogram by experimental evidence?

This is like asking "how was it determined that pi was between 3.14 and 3.15 by experimental evidence"? You settle a mathematical question using mathematics, not experiments, unless you have no other choice.

> How about the sentence "Concordance here is a POSITIVE phenomenon which reflects the extent to which combinations of notes can elicit the same gestalt phenomena that apply to the study of single notes"?

It's an opinion. It could be argued for. If you want experimental evidence for or against, you'd first need to decide what would count as evidence confirming or refuting it. I prefer not to get into that sort of thing myself, but if you want to argue it's wrong you should state your own psychoacoustic claims, and if you want to argue it's nonsense then you should make that case.

> How about "the pythagorean comma 531441:524288"?
> I assure you there is zero experimental support for every last decimal there, this is
> pure fantasy that this has any significance.

Sorry, but speaking as number theorist I assure you every digit is significant if you form a ratio of integers such as 3^12/2^19. There's no point in arguing against the fundamental theorem of arithmetic, which says such a ratio cannot possibly be equal to 1, no matter how close it comes. Speaking as a music theorist I can tell you that the fact it is approximately 1 is significant. This is completely basic, and if you don't know that, you should learn the basics before attempting a critique. Ratios which are close to but not exactly 1--ratios such as 81/80, 225/224 or 32805/32768--are more than mere numerical curiosities or things which can be ignored on the grounds that they are small. They entail a whole suite of consequences, and that fact you need to know to understand this business and be in any kind of position to give an intelligent critique of anything.

I thought your education was in mathematics? How can any mathematician speak of "zero experimental support" for the digits of 3^12/2^19? What does that even *mean*?

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:

> > One of the things you should have learned before writing this
> >
> > http://dl.dropbox.com/u/3507527/MusicTh.html
> >
> > is how to distinguish Paul's papers from these mystical
> > manuscripts.
>
>
> I don't see the necessity to discredit Warren's entire hypothesis
> as worthless simply because he doesn't understand the value in
> some of Paul's ideas. This is a very snarky remark on your
> behalf Carl, if it isn't already dull to say the least.

I returned comments in kind, but unlike Warren, I actually
read the document in question. And unlike Paul's document,
Warren's *is* the one more likely to yield to criticisms
of the kind he's throwing around (lack of experimental
evidence, vagueness, etc).

-Carl

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> I thought your education was in mathematics? How can any mathematician speak of "zero experimental support" for the digits of 3^12/2^19? What does that even *mean*?

By the way, were you not the fellow who recently posted Stormer pairs up to absurdly high values? If you are going to post commas which are beyond minuscule yourself, cavailing at the classic and long-recognized Pythagorean comma seems to involve you in ironic self-contradiction.

http://en.wikipedia.org/wiki/Pythagorean_comma

Gene wrote:

> > How about the sentence "Concordance here is a POSITIVE
> > phenomenon which reflects the extent to which combinations of
> > notes can elicit the same gestalt phenomena that apply to the
> > study of single notes"?
>
> It's an opinion. It could be argued for.

Actually it's a definition, of "concordance". The synthesis
of multiple partials into "gestalt" pitches is one of the most
studied in psychoacoustics. Paul's just defining concordance
(as many authors have done) as a higher-order manifestation of
this. It's useful to have such a working definition, but the
conclusions of the paper, like the ones I made in a post here
earlier today, in no way depend on it.

> > How about "the pythagorean comma 531441:524288"?
> > I assure you there is zero experimental support for every
> > last decimal there, this is pure fantasy that this has any
> > significance.
[snip]
> I thought your education was in mathematics? How can any
> mathematician speak of "zero experimental support" for the
> digits of 3^12/2^19? What does that even *mean*?

It means he has a B.S. detector tuned to the word "pythagorean",
not realizing that it is simply the name of an important
comma that has been the basis of tuning practice since the
Baroque (at least).

-Carl

> Posted by: "Carl Lumma" carl@lumma.org  
> <mailto:carl@lumma.org?Subject=%20Re%3A%20Optimal%20octave%20stretching%2Fcompressing>
> clumma <http://profiles.yahoo.com/clumma>
> Thu Mar 15, 2012 11:31 am (PDT)
>
>
> Hi Scott,
>
> The question of optimal octave stretch has been studied here.
> We have methods to compute optimal stretch for any regular
> temperament... not just EDOs, but also higher-rank temperaments
> represented by MOS scales (aka well-formed scales) and so on.
> The two most popular of these methods are called TOP tuning
> and TE tuning. They are very similar and give similar results.
>
> TOP is lucidly explained by Paul Erlich, who discovered it
> http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf
Funny. I'd hours before come across that exact paper (without really knowing what it was about), and put it in my "reading queue". Thanks for the recommendations.
Also, great flame war, guys...

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> I don't like the "professionals can be trained to appreciate it" standard, I prefer the "uninitiated ordinary people prefer it" standard.
> Because with the former standard, virtually anything perceivable would qualify.

I mostly agree with this. The only valid rebuttal I can think of is that most people are not "uninitiated", but have been exposed to music based on 12edo and syntonic temperament their whole lives and are thoroughly used to it. You can debate about how strong that rebuttal is, but I still think you have a good point about "sounding good to ordinary people".

> Also, I'm interested in what music reveals about the nature of the human mind.
>
> Re some other stuff, I did read Erlich and you should stop complaining about that,
> otherwise I'll say how the fact you haven't read Lehrdahl+Jackendoff, which was clearly the premier work in the field, proves you are an utter incompetent. (Actually I do not believe that, I'm just saying if that's the road you want to take, then that's where you'd be.)

Yeah, I agree, some people have been really uncivil toward Warren here.

BTW it's "Lerdahl" not "Lehrdahl", and the book is here http://cognet.mit.edu/library/books/view?isbn=026262107X available electronically through my university library (perhaps yours too). It appears to contain none of the words "tuning", "temperament", or "intonation", but it does mention "overtones" several times.

Keenan

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > I thought your education was in mathematics? How can any mathematician speak of "zero experimental support" for the digits of 3^12/2^19? What does that even *mean*?
>
> By the way, were you not the fellow who recently posted Stormer pairs up to absurdly high values? If you are going to post commas which are beyond minuscule yourself, cavailing at the classic and long-recognized Pythagorean comma seems to involve you in ironic self-contradiction.
>
> http://en.wikipedia.org/wiki/Pythagorean_comma

Let's excuse Warren for not recognizing this relatively famous rational number in our circles (and especially remain civil about it).

(But in all fairness, Warren, your complaints about the Middle Path paper are unfounded, as others have already pointed out. It's certainly not written in the style of a mathematical or scientific paper, but once you get past the language you may see that it introduced a few extremely important ideas.)

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Let's excuse Warren for not recognizing this relatively famous rational number in our circles (and especially remain civil about it).

I thought "ironic self-contradiction" was well within the boundaries of civility originally set by "reads vaguely like old mystical manuscripts", but I agree the discussion was handled badly and regret any role I may have played in that. Warren is certainly equipped to give a critique, but we didn't get one because he didn't actually read Paul's paper well enough to understand what it said. I had hoped to prod him into doing so.

Gene wrote:

>I thought "ironic self-contradiction" was well within the boundaries
>of civility originally set by "reads vaguely like old mystical
>manuscripts", but I agree the discussion was handled badly and regret
>any role I may have played in that. Warren is certainly equipped to
>give a critique, but we didn't get one because he didn't actually read
>Paul's paper well enough to understand what it said. I had hoped to
>prod him into doing so.

Warren panned Paul's paper without reading it, and when I provided
other links and discussion, he ignored that and threw more flames.
He's been a member here long enough, and enough resources are
available on wikipedia and elsewhere, that this should not have been
an issue. I feel I responded fairly and I frankly doubt my
detractors have even bothered to read my posts in this thread. Ample
opportunity for on-topic discussion was available therein. None of
Warren's messages were touched and no users were "placed on moderation".
Complaints about moderator policy can be addressed offlist. Complaints
onlist will be deleted. Ryan: your e-mail address is bouncing.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Gene wrote:
>
> >I thought "ironic self-contradiction" was well within the boundaries
> >of civility originally set by "reads vaguely like old mystical
> >manuscripts", but I agree the discussion was handled badly and regret
> >any role I may have played in that. Warren is certainly equipped to
> >give a critique, but we didn't get one because he didn't actually read
> >Paul's paper well enough to understand what it said. I had hoped to
> >prod him into doing so.
>
> Warren panned Paul's paper without reading it...

What I meant by "uncivil" was that, if somebody claims they read something, and then makes comments about it that you disagree with, it's uncivil to say "no, you must be lying, you wouldn't say that if you had actually read it".

Saying "I strongly disagree with that comment and urge you to read the paper more carefully and think again!" is civil, but saying "don't criticize things without actually reading them" is uncivil because it implies they were lying about having read it. Let's assume good faith here.

Keenan

Keenan wrote:

>What I meant by "uncivil" was that, if somebody claims they read
>something, and then makes comments about it that you disagree with,
>it's uncivil to say "no, you must be lying, you wouldn't say that if
>you had actually read it".

He admitted several times that he didn't read it!

>Saying "I strongly disagree with that comment and urge you to read the
>paper more carefully and think again!" is civil, but saying "don't
>criticize things without actually reading them" is uncivil because it
>implies they were lying about having read it. Let's assume good faith here.

I reiterate my claim that you haven't read the thread.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Keenan wrote:
>
> >What I meant by "uncivil" was that, if somebody claims they read
> >something, and then makes comments about it that you disagree with,
> >it's uncivil to say "no, you must be lying, you wouldn't say that if
> >you had actually read it".
>
> He admitted several times that he didn't read it!

Huh? I was going by this thing Warren said, that I saw as I was reading the thread:

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
> Re some other stuff, I did read Erlich and you should stop complaining about that,

> >Saying "I strongly disagree with that comment and urge you to read the
> >paper more carefully and think again!" is civil, but saying "don't
> >criticize things without actually reading them" is uncivil because it
> >implies they were lying about having read it. Let's assume good faith here.
>
> I reiterate my claim that you haven't read the thread.

I did read the thread! And I didn't see anywhere where Warren said he didn't read Paul's paper. In fact, I'm searching now and still don't see it.

Have we all been taking crazy pills??

Keenan

On 3/15/2012 5:23 PM, Mike Battaglia wrote:

> Keenan Pepper's done some work on finding scales that minimize HE. The
> best one for harmony overall ended up being meantone, which has a
> generator of half a meantone fourth. Unfortunately, I don't care much
> about this because it means that Westerners ended up already finding
> "the best scale." Well, what's "best" to me is finding something
> -new-.

I think that's a good point in that what many of us are looking for is not some idealized musical scale, but rather something that sounds different from traditional music theory. A scale with its own internal structure and logic that allows for new kinds of music. Some of the scales that are the most interesting to me have been ones that don't rate the best in the typical "badness" evaluations we use in the regular mapping paradigm, but they're not far below the surface (e.g. superpelog[14], lemba[16]).

It really is hard to beat meantone, though, and even some alternatives like injera and mohajira have a close relationship with meantone.

> Anyway, the problem is that these metrics don't adequately represent,
> to me, all of the things I like about a tuning system or a scale. If
> they did, I wouldn't still be here. There are other scales with crap
> for harmony that have the most beautiful logical, organizational,
> "conceptual," "functional," whatever-you-want layouts I've ever seen,
> and it's a worthwhile tradeoff for me in those systems to accept some
> beating in the harmony. So if what you want is more relevant things to
> model, and more "finders" to find the best tunings that have those
> things - then yes, I agree. So what do we model?

Suitability for generalized keyboards, for one thing. One of the reasons I have an interest in orwell (although I generally prefer less complex temperaments) is how well it fits on a generalized keyboard, even when extended up to the 13-limit. But short of trying a number of different options by hand on a QWERTY keyboard for each temperament, I don't have a good way to test these properties.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> On 3/15/2012 5:23 PM, Mike Battaglia wrote:
>
> > Keenan Pepper's done some work on finding scales that minimize HE. The
> > best one for harmony overall ended up being meantone, which has a
> > generator of half a meantone fourth.

> It really is hard to beat meantone, though

Especially if you define it so that it includes half a meantone fourth as a generator.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@> wrote:
> >
> > On 3/15/2012 5:23 PM, Mike Battaglia wrote:
> >
> > > Keenan Pepper's done some work on finding scales that minimize HE. The
> > > best one for harmony overall ended up being meantone, which has a
> > > generator of half a meantone fourth.
>
> > It really is hard to beat meantone, though
>
> Especially if you define it so that it includes half a meantone fourth as a generator.

That was a typo; I meant to say "ended up being meantone, and semaphore, which has a generator of half a meantone fourth." Whoops.

-Mike

hi warren,

sorry to see you taking such a beating here ... but it was
quite silly of you to refer to "decimal points" in the
ratio of the pythagorean-comma.

anyway ... i used a spreadsheet years ago to calculate
DES scales for many of the TOP temperaments in Erlich's paper,
and then made Tonescape .tuning files of them, and also
.tonescape musical-piece files illustrating the scales
and some chords.

i thought i had posted these in a "Files" section of one
of the Yahoo groups, but can't find them anywhere ... i guess
i must have been waiting to finish the whole set before i
posted them. anyway, i'll post a few so that you and others
can play around with them in Tonescape.

-monz
http:/tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@> wrote:
> >
> > > If you really can't see a single thing in Erlich's paper that might have stimulated my "strange" reaction then I assure you that me pointing one out, would not change that.
> >
> > Without a single specific example and critique this is just bullshit, and you are intelligent enough to produce something of actual substance instead. I'm asking, again, but I will call bullshit on bullshit.
>
>
> --ok, since you absolutely insist, how was (every decimal of) the number "1089.38" justified in the TOP Wurschmidt horogram by experimental evidence?
>
> How about the sentence "Concordance here is a POSITIVE phenomenon which reflects the extent to which combinations of notes can elicit the same gestalt phenomena that apply to the study of single notes"?
>
> How about "the pythagorean comma 531441:524288"?
> I assure you there is zero experimental support for every last decimal there, this is
> pure fantasy that this has any significance.
>
> [Note: I got these 3 examples by tossing a dart at the paper 8 times and picking three of the 8 impact sites. They have no significance aside from that.]
>

ah, found them! ... they are in the Yahoo tuning_files group:

/tuning-math/files/monz/tonescape-top/

(delete line-break in URL if necessary)

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> hi warren,
>
> sorry to see you taking such a beating here ... but it was
> quite silly of you to refer to "decimal points" in the
> ratio of the pythagorean-comma.
>
> anyway ... i used a spreadsheet years ago to calculate
> DES scales for many of the TOP temperaments in Erlich's paper,
> and then made Tonescape .tuning files of them, and also
> .tonescape musical-piece files illustrating the scales
> and some chords.
>
> i thought i had posted these in a "Files" section of one
> of the Yahoo groups, but can't find them anywhere ... i guess
> i must have been waiting to finish the whole set before i
> posted them. anyway, i'll post a few so that you and others
> can play around with them in Tonescape.
>
> -monz
> http:/tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>
> --- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > >
> > >
> > >
> > > --- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@> wrote:
> > >
> > > > If you really can't see a single thing in Erlich's paper that might have stimulated my "strange" reaction then I assure you that me pointing one out, would not change that.
> > >
> > > Without a single specific example and critique this is just bullshit, and you are intelligent enough to produce something of actual substance instead. I'm asking, again, but I will call bullshit on bullshit.
> >
> >
> > --ok, since you absolutely insist, how was (every decimal of) the number "1089.38" justified in the TOP Wurschmidt horogram by experimental evidence?
> >
> > How about the sentence "Concordance here is a POSITIVE phenomenon which reflects the extent to which combinations of notes can elicit the same gestalt phenomena that apply to the study of single notes"?
> >
> > How about "the pythagorean comma 531441:524288"?
> > I assure you there is zero experimental support for every last decimal there, this is
> > pure fantasy that this has any significance.
> >
> > [Note: I got these 3 examples by tossing a dart at the paper 8 times and picking three of the 8 impact sites. They have no significance aside from that.]
> >
>

--- In tuning-math@yahoogroups.com, Scott Nordlund <gsn10@...> wrote:
>
> There's no reason for me to go any further down that road
> if it's already been perfected by someone else.
>
> Should I keep working on it,
> or is it considered a solved problem already?

hi Scott,
here some general information and further links about that topic:

http://en.wikipedia.org/wiki/Pseudo-octave#Stretched_octave
http://en.wikipedia.org/wiki/Stretched_tuning

Especially for pianos an adaequate choice depends
mostly on the given inharmonicity of strings
http://www.pianoworld.com/forum/ubbthreads.php/ubb/showflat/Forum/3/topic/004285/Number/0/site_id/1
Quote:
"
The first consideration in choosing how much stretch is needed, must be the piano size. Large grand pianos have very long strings in the bass and the partial tones have minimal sharpening. Generally the amount of stretch will be lower for this frame. Small pianos have short heavy strings and very sharp partials and thus will require much more stretch to sound in tune. The next consideration is the kind of musical performance expected. For intimate chamber music, an excessive stretch will make it hard to blend the sounds with good intonation with other instruments. But on the solo concert stage, more stretch is needed to project with brilliant highs and full bass.
"

Conclusion:
Hence there exist no "optimal" stretch
even for one single individual piano
for all musical purposes,
not to mention the personal human listeners
preference, depending on habituation
and personal taste of the tuner.

For more psycho-acousic concepts
see finally the specific literature in:
http://www.mmk.ei.tum.de/persons/ter/top/octstretch.html

Here attend the paper:
https://digitalcollections.anu.edu.au/bitstream/.../2/Bell_ICMPC7.pdf
on page 3, that refers to the devition out of 2/1:
"
The octave is not exactly 1200cents,
but streched by 13cents. there are a number of explantions of why psychophysical judements of the octave always exceed the expeted 2:1 ratio.... Ward [7] prefers ~1210cents"

more en detail:
http://books.google.de/books?id=sE1mk8ed1dkC&pg=PA102&lpg=PA102&dq=Ward%27s+curve+of+octave+stretching&source=bl&ots=2Oqr_e_8cS&sig=6XMxQ0lm1dVXgp1LIWJGsX7Cga4&hl=de&sa=X&ei=RJRnT4mVIabR4QSH-oCLCQ&ved=0CCYQ6AEwAA#v=onepage&q=Ward%27s%20curve%20of%20octave%20stretching&f=false

There study the empirically plot in figure 29 on page 129
about
http://www.columbia.edu/~zsp2101/octaves/octaves.pdf

-Andy

For those that care:

This is the best formula for the inharmononicity of the n-th partial that I
know. This works on any iron wire instrument that are struck or plucked.
Partial are harmonic on bowed instruments (tell the bow is lifted). Might
add, organ pipe partial are harmonic tell air is stopped then they go sharp.

In = 1731 (n^2 - 1)*(S*(1 + B/8) + (3*B/(1 + B))*(abs(a/L – (S)^0.5)^3 +
abs(b/L – (S)^0.5)^3))

S = d^4/139430*L^2*T

B = A*(D^2/d^2 – 1)

L = speaking length (in)

D = steel wire diameter (mils)

D = overall string diameter (mils)

T = string tension (lbs)

a = agraffe-to-start-of-winging (in)

b = bridge-pinto-start-of winding (in)

n = partial number (1 = fundamental)

A = 0.89 for copper wrapped

= 0.79 for iron

= 0.27 for aluminum

= 0 for no wrapping

If you think I made a mistake let me know and I will check it again;
getting then ( ) rite is hard for me.

ƒg

On Mon, Mar 19, 2012 at 4:22 PM, Andy <a_sparschuh@yahoo.com> wrote:

> **
>
>
> --- In tuning-math@yahoogroups.com, Scott Nordlund <gsn10@...> wrote:
> >
> > There's no reason for me to go any further down that road
> > if it's already been perfected by someone else.
> >
> > Should I keep working on it,
> > or is it considered a solved problem already?
>
> hi Scott,
> here some general information and further links about that topic:
>
> http://en.wikipedia.org/wiki/Pseudo-octave#Stretched_octave
> http://en.wikipedia.org/wiki/Stretched_tuning
>
> Especially for pianos an adaequate choice depends
> mostly on the given inharmonicity of strings
>
> http://www.pianoworld.com/forum/ubbthreads.php/ubb/showflat/Forum/3/topic/004285/Number/0/site_id/1
> Quote:
> "
> The first consideration in choosing how much stretch is needed, must be
> the piano size. Large grand pianos have very long strings in the bass and
> the partial tones have minimal sharpening. Generally the amount of stretch
> will be lower for this frame. Small pianos have short heavy strings and
> very sharp partials and thus will require much more stretch to sound in
> tune. The next consideration is the kind of musical performance expected.
> For intimate chamber music, an excessive stretch will make it hard to blend
> the sounds with good intonation with other instruments. But on the solo
> concert stage, more stretch is needed to project with brilliant highs and
> full bass.
> "
>
> Conclusion:
> Hence there exist no "optimal" stretch
> even for one single individual piano
> for all musical purposes,
> not to mention the personal human listeners
> preference, depending on habituation
> and personal taste of the tuner.
>
> For more psycho-acousic concepts
> see finally the specific literature in:
> http://www.mmk.ei.tum.de/persons/ter/top/octstretch.html
>
> Here attend the paper:
> https://digitalcollections.anu.edu.au/bitstream/.../2/Bell_ICMPC7.pdf
> on page 3, that refers to the devition out of 2/1:
> "
> The octave is not exactly 1200cents,
> but streched by 13cents. there are a number of explantions of why
> psychophysical judements of the octave always exceed the expeted 2:1
> ratio.... Ward [7] prefers ~1210cents"
>
> more en detail:
>
> http://books.google.de/books?id=sE1mk8ed1dkC&pg=PA102&lpg=PA102&dq=Ward%27s+curve+of+octave+stretching&source=bl&ots=2Oqr_e_8cS&sig=6XMxQ0lm1dVXgp1LIWJGsX7Cga4&hl=de&sa=X&ei=RJRnT4mVIabR4QSH-oCLCQ&ved=0CCYQ6AEwAA#v=onepage&q=Ward%27s%20curve%20of%20octave%20stretching&f=false
>
> There study the empirically plot in figure 29 on page 129
> about
> http://www.columbia.edu/~zsp2101/octaves/octaves.pdf
>
> -Andy
>
>
>

I see that I missed this In is in ¢

On Mon, Mar 19, 2012 at 6:23 PM, Freeman Gilmore
<freeman.gilmore@gmail.com>wrote:

> For those that care:
>
>
>
> This is the best formula for the inharmononicity of the n-th partial that
> I know. This works on any iron wire instrument that are struck or
> plucked. Partial are harmonic on bowed instruments (tell the bow is
> lifted). Might add, organ pipe partial are harmonic tell air is stopped
> then they go sharp.
>
>
>
> In = 1731 (n^2 - 1)*(S*(1 + B/8) + (3*B/(1 + B))*(abs(a/L – (S)^0.5)^3 +
> abs(b/L – (S)^0.5)^3))
>
>
>
> S = d^4/139430*L^2*T
>
>
>
> B = A*(D^2/d^2 – 1)
>
>
>
> L = speaking length (in)
>
>
>
> D = steel wire diameter (mils)
>
>
>
> D = overall string diameter (mils)
>
>
>
> T = string tension (lbs)
>
>
>
> a = agraffe-to-start-of-winging (in)
>
>
>
> b = bridge-pinto-start-of winding (in)
>
>
>
> n = partial number (1 = fundamental)
>
>
>
> A = 0.89 for copper wrapped
>
> = 0.79 for iron
>
> = 0.27 for aluminum
>
> = 0 for no wrapping
>
>
> If you think I made a mistake let me know and I will check it again;
> getting then ( ) rite is hard for me.
>
> ƒg
>
>
>
>
>
>
>
>
>
>

On Thu, Mar 15, 2012 at 1:22 PM, Scott Nordlund <gsn10@hotmail.com> wrote:
>
> I’m guessing someone has done this already. I’m interested in alternate tunings, but not so well connected to the community.
>
>
> Since stretched octaves are well tolerated, or even preferred, and since some non-octave tunings have advantages over similar EDO tunings, is there, for a given equal temperament, an optimal way to stretch or compress the octave to minimize overall dissonance?
>
>
> You could imagine in 15-EDO, for example, some compromise between 1200 cent octaves (15-EDO with bad fifths) and 1170 cent octaves (Wendy Carlos alpha scale with bad octaves). This is an extreme example, but probably any equal temperament could be improved by stretching or compressing the octaves a little.

For instance, the tuning for 15-EDO that minimizes RMS weighted prime error is:

http://x31eq.com/cgi-bin/rt.cgi?ets=15&limit=5

It has an 1194-cent octave stretch.

-Mike