A dyadic analysis of EDOs 4 to 24

Igs, you should find this interesting. Here's an analysis of EDOs 4 to 24 showing which good harmony intervals occur in each EDO within +/-6.776 cents (256/255) accuracy. The harmony intervals that I consider to be good (less than an octave wide) are...

9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 10/7, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 11/6 and 13/7.

EDO4...none

EDO5...none

EDO6...9/8

EDO7...none

EDO8...11/6

EDO9...7/6, 12/7, 13/7

EDO10...none

EDO11...9/7, 11/8

EDO12...9/8, 4/3, 3/2

EDO13...11/8, 9/5

EDO14...9/7

EDO15...6/5, 5/3

EDO16...8/7, 7/4, 11/6

EDO17...4/3, 3/2, 11/7

EDO18...9/8, 7/6, 12/7, 13/7

EDO19...6/5, 5/3, 13/7

EDO20...11/7, 9/5

EDO21...8/7, 7/4

EDO22...7/6, 5/4, 9/7, 11/8, 8/5, 12/7

EDO23...9/8, 7/6, 6/5, 11/7, 5/3, 12/7, 11/6

EDO24...9/8, 4/3, 11/8, 3/2, 11/6

It looks like 23EDO is the most versatile with seven good intervals occurring which is strange because I've rarely seen 23EDO mentioned.

I don't bother with 25 or higher EDOs because they contain intervals narrower than 48.18855 cents which I consider to be illegal (see message number 96914 "Crows").

John.

But you can use a >25EDO and just choose the tones of the scale such that 2 tones are never 1 step of 25EDO apart.

Also, illegal can be wonderful ;) The small inflections in Indian Classical music are beautiful and illegal.

Mat Cooper

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> I don't bother with 25 or higher EDOs because they contain intervals narrower than 48.18855 cents which I consider to be illegal

Mat>>"But you can use a >25EDO and just choose the tones of the scale such that 2 tones are never 1 step of 25EDO apart."

I had thought of that and it would work. In light of this I'm going to have another look at 25 and higher EDOs.

John.

--- In tuning@yahoogroups.com, "ixlramp" <ixlramp@...> wrote:
>
> But you can use a >25EDO and just choose the tones of the scale such that 2 tones are never 1 step of 25EDO apart.
>
> Also, illegal can be wonderful ;) The small inflections in Indian Classical music are beautiful and illegal.
>
> Mat Cooper
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> > I don't bother with 25 or higher EDOs because they contain intervals narrower than 48.18855 cents which I consider to be illegal
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> EDO14...9/7

The least number of "good intervals" for any edo 11 or above.

> EDO17...4/3, 3/2, 11/7

Generates the 2.3.11/7 subgroup, so you get 1-14/11-3/2 triads.

> EDO18...9/8, 7/6, 12/7, 13/7

The 2.9.21.39 subgroup.

> EDO23...9/8, 7/6, 6/5, 11/7, 5/3, 12/7, 11/6

The 2.9.15.21.33 subgroup.

> EDO24...9/8, 4/3, 11/8, 3/2, 11/6

The 2.3.11 subgroup.

> It looks like 23EDO is the most versatile with seven good intervals occurring which is strange because I've rarely seen 23EDO mentioned.

It does have some possibilities.

My philosophy is this ...

There are no good or bad intervals.
Nothing is illegal, everything is permissible.

**********************************************

72EDO has the amazing property of approximating every simple 11 limit interval to within 4 cents.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> I had thought of that and it would work. In light of this I'm going to have another look at 25 and higher EDOs.

Mat Cooper

Hi John,

I did this same task a while ago, except that I included intervals in the 2nd octave as well, and used slightly more tolerance in what I consider "good", such that I would add the following intervals to your list:
3/1, 5/2, 7/2, 7/3, 9/4, 10/3, 11/3, 11/4, 11/5, 11/9, 12/5, 13/4, 13/5, 13/6, and 15/4 (I may have forgotten a few). This definitely alters the gamut. Also, I allowed a broader tolerance of up to 8 or 9 cents (nothing too rigorously-defined, as different ratios seem to vary in their sensitivity to mistuning). If you like I can post what I came up with, but it will also be available in my upcoming e-book.

>
It looks like 23EDO is the most versatile with seven good intervals occurring which is strange because I've rarely seen 23EDO mentioned.
>

Probably because one can't play common-practice music in 23. I love it, though. It has a lot of good possibilities. It's excellent for music based on 5:6:7 or 3:5:7 harmonic triads. Someday I might get a guitar fretted up to it, though 23's cutting it a bit narrow.

>
I don't bother with 25 or higher EDOs because they contain intervals narrower than 48.18855 cents which I consider to be illegal (see message number 96914 "Crows").
>

I'd probably draw the line at 30 rather than 25. I've written some music in 25 where the 48-cent interval is used as a semitone(!) melodically, and it works surprisingly well. At 40 cents though things are definitely breaking down, sounding more like a quaver and less like a step.

-Igs

On Sat, Mar 19, 2011 at 11:31 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Hi John,
>
> I did this same task a while ago, except that I included intervals in the 2nd octave as well, and used slightly more tolerance in what I consider "good", such that I would add the following intervals to your list:
> 3/1, 5/2, 7/2, 7/3, 9/4, 10/3, 11/3, 11/4, 11/5, 11/9, 12/5, 13/4, 13/5, 13/6, and 15/4 (I may have forgotten a few). This definitely alters the gamut. Also, I allowed a broader tolerance of up to 8 or 9 cents (nothing too rigorously-defined, as different ratios seem to vary in their sensitivity to mistuning). If you like I can post what I came up with, but it will also be available in my upcoming e-book.

How upcoming is it?

I'd like to add that I'd be more open to all of the output from these
alternative theories, John's and Michael's and everyone else's, if
people would start talking more about triads than dyads. Or preferably
triads with the root being doubled down an octave, which I guess are
tetrads. I just don't care about dyadic music, and you can take a lot
of "good" dyads and smush them into a chord that sounds terrible.

-Mike

Although you may not consider it valid for 3+-ads, John does at least
explain his working theory for those chords in his book, which fits into the
same framework (and explains some of what was for me confusing terminology
about independent note "strength").

On Sat, Mar 19, 2011 at 11:53 PM, Daniel Nielsen <nielsed@...> wrote:
>
> Although you may not consider it valid for 3+-ads, John does at least explain his working theory for those chords in his book, which fits into the same framework (and explains some of what was for me confusing terminology about independent note "strength").

As I've said before, I think that John's ideas are somewhat on the
right track. His equation is, roughly, a weighted average of critical
band roughness and Tenney Height, which are both types of dissonance.

I don't like the arbitrary 256/255 cutoff, and I don't like that he's
just picked one way to average these and decided that it's "good," and
I don't like that he's using Tenney Height instead of HE for the
concordance part of the equation. I also don't like that not a lot of
explanatory power has been given here and that it's presented as a
simple "master equation," but other than that it isn't too bad.

But you still can't just take dyads together and smush them randomly
together into chords and write consonant music from it.

-Mike

I don't know, Mike B, there can be simple "rules", it seems to me, for
making harmonic chords from consonant intervals. Now that doesn't mean that
that edo has the same "rules", since it relies on a different tonal
structure. I have questioned John about his "kiss" interval, and he doesn't
seem to have an explanation except that it sounds right to him and it
happens to be in keeping with his use of pow2. That's not bad, though, that
he is using his ears to try to keep his model valid.

One thing I like about John's model is that the evaluation is essentially a
surprisal value (when considered probabilistically). Granted, by this
interpretation we have to allow "probabilities" that exceed 1. If we rewrite
it as

-lb(2+1/x+1/y-diss(x,y))

then, according to John's classification,

value > 3 intolerable
else > 2 tolerable but not sweet
else > 1 ultra minor
else > 0 blue minor
else major

In this way inharmonicity might be interpreted as "surprise". At a certain
threshold, one might expect that the surprise is no longer surprising and
falls into the category of noise. Given a probabilistic interpretation, it
becomes unlikely that one intended such an exact outcome, and other
hypotheses become more probable. That was one notion, anyway.

On Sun, Mar 20, 2011 at 12:40 AM, Daniel Nielsen <nielsed@...> wrote:
>
> I don't know, Mike B, there can be simple "rules", it seems to me, for making harmonic chords from consonant intervals.

I don't see why you say this. Tenney height works decently well for
triads and is the reason for the recent interest in 10:13:15 around
here. If you find Tenney Height doesn't suit your needs, triadic HE
takes its place. If John wants to come up with an overall "goodness"
measure for triads that weights the consonance from various
psychoacoustic factors, he certainly can (and I think that he has
previously). Ease of F0 estimation, critical band interactions, and
"tonalness" (if it's different from the first criterion) are all
things that could be weighted in here.

> Now that doesn't mean that that edo has the same "rules", since it relies on a different tonal structure. I have questioned John about his "kiss" interval, and he doesn't seem to have an explanation except that it sounds right to him and it happens to be in keeping with his use of pow2. That's not bad, though, that he is using his ears to try to keep his model valid.

One is allowed to come up with any model one wants. I don't remember
which one the "kiss" interval is (was it 256/255?), but assuming that
it turns out to be valid in listening tests for a wide range of
people, it will be valid. Given how well HE has held up, which assumes
a Gaussian-sized kiss "range," I would be surprised if it does. All I
want above is to perhaps provide an expositional amount of
psychoacoustic explanation for his model so as to encourage him to
research it further.

> One thing I like about John's model is that the evaluation is essentially a surprisal value (when considered probabilistically).

The information theory word for surprisal is entropy. So if you're
into this interpretation of John's model, then you should probably
look into harmonic entropy, which aims to measure this directly.

The main difference between this and HE is that John assumes a
rectangular mistuning range of 256/255 in either direction, where
anything that's in the range is "good" and anything outside of it is
"bad."

HE assumes a Gaussian-tapered range with a variable standard deviation
(usually 1-1.2%) and is deliberately set up so that the probabilities
overlap. The more

> Granted, by this interpretation we have to allow "probabilities" that exceed 1. If we rewrite it as
>
> -lb(2+1/x+1/y-diss(x,y))

What does lb mean here?

> then, according to John's classification,
>
> value > 3 intolerable
> else > 2 tolerable but not sweet
> else > 1 ultra minor
> else > 0 blue minor
> else major
> In this way inharmonicity might be interpreted as "surprise".

Indeed, but I think that HE does the job better.

> At a certain threshold, one might expect that the surprise is no longer surprising and falls into the category of noise. Given a probabilistic interpretation, it becomes unlikely that one intended such an exact outcome, and other hypotheses become more probable. That was one notion, anyway.

That may be related to this:

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

-Mike

On Sun, Mar 20, 2011 at 1:12 AM, Mike Battaglia <battaglia01@...> wrote:
>
> The main difference between this and HE is that John assumes a
> rectangular mistuning range of 256/255 in either direction, where
> anything that's in the range is "good" and anything outside of it is
> "bad."
>
> HE assumes a Gaussian-tapered range with a variable standard deviation
> (usually 1-1.2%) and is deliberately set up so that the probabilities
> overlap. The more

Uh, I forgot to finish this sentence.

Roughly speaking, HE assumes a Gaussian-tapered range with a variable
standard deviation (usually 1-1.2%) and is deliberately set up so that
the ranges for each interval overlap. These intervals are assumed to
be the basis vectors for a harmonic signal (which is one criticism I
have of the model). The more intervals that an interval "activates,"
the more information the signal carries, hence the more surprising it
is, hence the higher in entropy it is.

-Mike

>
> I don't see why you say this. Tenney height works decently well for
> triads and is the reason for the recent interest in 10:13:15 around
> here. If you find Tenney Height doesn't suit your needs, triadic HE
> takes its place. If John wants to come up with an overall "goodness"
> measure for triads that weights the consonance from various
> psychoacoustic factors, he certainly can (and I think that he has
> previously). Ease of F0 estimation, critical band interactions, and
> "tonalness" (if it's different from the first criterion) are all
> things that could be weighted in here.
>

Really I just like the look and feel of John's formula, and I tend to trust
mathematical elegance. I think such a formula would be a useful tool, and
that it could even reveal a pragmatic psychophysical truth. I am just
starting down my microtonal study, but I wonder if Tenney height might rate
certain (few) intervals a little to highly.

I definitely think John's model should be adapted to basically reflect
reality where needed, however.

One is allowed to come up with any model one wants. I don't remember
> which one the "kiss" interval is (was it 256/255?), but assuming that
> it turns out to be valid in listening tests for a wide range of
> people, it will be valid. Given how well HE has held up, which assumes
> a Gaussian-sized kiss "range," I would be surprised if it does. All I
> want above is to perhaps provide an expositional amount of
> psychoacoustic explanation for his model so as to encourage him to
> research it further.
>

It would be interesting to see or make a comparison of his model to HE
certainly.

The information theory word for surprisal is entropy. So if you're
> into this interpretation of John's model, then you should probably
> look into harmonic entropy, which aims to measure this directly.
>

Although there are historically many somewhat inconsistent terms and
interpretations when it comes to information, entropy, and symmetry, the
only other names I know surprisal to go by are self-information, information
value, and bit width of a fraction.
I do notice that Wikipedia states "The information
entropy<http://en.wikipedia.org/wiki/Information_entropy>of a random
event is the expected
value <http://en.wikipedia.org/wiki/Expected_value> of its
self-information". Is this what you mean?

I definitely want to look into harmonic entropy when I get the chance.
Honestly, though, I may prefer simply to review the data that is produced by
such methods.

The main difference between this and HE is that John assumes a
> rectangular mistuning range of 256/255 in either direction, where
> anything that's in the range is "good" and anything outside of it is
> "bad."
>
> HE assumes a Gaussian-tapered range with a variable standard deviation
> (usually 1-1.2%) and is deliberately set up so that the probabilities
> overlap. The more intervals that an interval "activates,"
> the more information the signal carries, hence the more surprising it
> is, hence the higher in entropy it is.
>

That's good, but is it elegant? I can punch out John's formula on my
calculator for any set of notes. Once again, that doesn't mean John's
methods are valid, but I tend to think a simple method may lead to more
significant discoveries concerning interpretation.

What does lb mean here?
>

lb() is the standardized way of writing log2(), which is sometimes also
written lg(). It painfully resembles "pound".

> > then, according to John's classification,
> >
> > value > 3 intolerable
> > else > 2 tolerable but not sweet
> > else > 1 ultra minor
> > else > 0 blue minor
> > else major
> > In this way inharmonicity might be interpreted as "surprise".
>
> Indeed, but I think that HE does the job better.
>

Depends on what is meant by better, but I won't argue that HE models well
the variables it measures. However, to me, that doesn't imply construct
validity or mean that it is the most effective way to make music. I would
have to tinker much more.

> That may be related to this:
>
> http://en.wikipedia.org/wiki/Cocktail_party_effect >
>

I'll have fun reading this in the morning.

On Sun, Mar 20, 2011 at 2:15 AM, Daniel Nielsen <nielsed@...> wrote:
>
>> I don't see why you say this. Tenney height works decently well for
>> triads and is the reason for the recent interest in 10:13:15 around
>> here. If you find Tenney Height doesn't suit your needs, triadic HE
>> takes its place. If John wants to come up with an overall "goodness"
>> measure for triads that weights the consonance from various
>> psychoacoustic factors, he certainly can (and I think that he has
>> previously). Ease of F0 estimation, critical band interactions, and
>> "tonalness" (if it's different from the first criterion) are all
>> things that could be weighted in here.
>
> Really I just like the look and feel of John's formula, and I tend to trust mathematical elegance.

Unfortunately, nothing about the workings of the brain or the inner
ear are mathematically elegant. This is partly why they're so hard to
model.

Many people have come onto these lists and decided that all music can
be explained by secret patterns in numbers. In contrast, other folks
have decided that things like psychoacoustics play more of a role.
Coming up with a mathematically elegant, quick "rule of thumb" that
approximates the behavior of more complex models is a worthy goal. I
don't think John claims that this is what he's doing, but rather that
his equation is a psychoacoustic model in and of itself.

> I think such a formula would be a useful tool, and that it could even reveal a pragmatic psychophysical truth. I am just starting down my microtonal study, but I wonder if Tenney height might rate certain (few) intervals a little to highly.

What's more important is to know the concepts behind each model.
Tenney height matches a lot of intervals a little too "lowly," namely
things like 500000/400001, which are indistinguishable from 5/4. But
the point of Tenney height is just to reflect that the consonance of
intervals tends to roughly follow an n*d pattern.

> I definitely think John's model should be adapted to basically reflect reality where needed, however.

You are allowed to be a fan of John's model if you want. Again, the
important thing is to understand the concepts at play here.

>> The information theory word for surprisal is entropy. So if you're
>> into this interpretation of John's model, then you should probably
>> look into harmonic entropy, which aims to measure this directly.
>
> Although there are historically many somewhat inconsistent terms and interpretations when it comes to information, entropy, and symmetry, the only other names I know surprisal to go by are self-information, information value, and bit width of a fraction.
>  I do notice that Wikipedia states "The information entropy of a random event is the expected value of its self-information". Is this what you mean?

Yes. Perhaps I misspoke. Looks like it's saying that entropy is the
expected value of self-information, and surprisal is self-information.
I was taught that entropy is self-information, which Wikipedia
acknowledges is a common usage of the word here:

"The term self-information is also sometimes used as a synonym of
entropy, i.e. the expected value of self-information in the first
sense, because I(X;X) = H(X), where I(X;X) is the mutual information
of X with itself.[1]"

> I definitely want to look into harmonic entropy when I get the chance. Honestly, though, I may prefer simply to review the data that is produced by such methods.

Here's a somewhat decent page on it:

http://sonic-arts.org/td/erlich/entropy.htm

Note that it doesn't represent critical band effects, but 60 cents
turns out to be pretty dissonant anyways for other reasons.

>> The main difference between this and HE is that John assumes a
>> rectangular mistuning range of 256/255 in either direction, where
>> anything that's in the range is "good" and anything outside of it is
>> "bad."
>>
>> HE assumes a Gaussian-tapered range with a variable standard deviation
>> (usually 1-1.2%) and is deliberately set up so that the probabilities
>> overlap. The more intervals that an interval "activates,"
>> the more information the signal carries, hence the more surprising it
>> is, hence the higher in entropy it is.
>
> That's good, but is it elegant? I can punch out John's formula on my calculator for any set of notes. Once again, that doesn't mean John's methods are valid, but I tend to think a simple method may lead to more significant discoveries concerning interpretation.

It's not elegant at all, but it's accurate. The Fourier Transform is
also not elegant, but is a staple of just about everything that we do
here. If you want to come up with an elegant, quick, shorthand way to
figure out how consonant an interval, some parameter averaging
critical band dissonance and ease of F0 estimation is probably a good
way to do it.

I don't think there's any mystery as to why his formula sometimes
"works," except for the holes that have already been mentioned, and I
don't think there should be any difficulty in interpreting it either.

Tenney height < 100 would probably work roughly as well.

>> What does lb mean here?
>
> lb() is the standardized way of writing log2(), which is sometimes also written lg(). It painfully resembles "pound".

>> Indeed, but I think that HE does the job better.
>
> Depends on what is meant by better, but I won't argue that HE models well the variables it measures. However, to me, that doesn't imply construct validity or mean that it is the most effective way to make music. I would have to tinker much more.

All of these criticisms apply a million times more to John's formula
than to harmonic entropy.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> How upcoming is it?

Well, I've almost finished the first draft of the "Microtonality 101" section, which is the "hard" part. Dealing with the EDOs will be straight-forward enough. Hopefully I won't decide to change horses mid-stream again, because I've re-written the damn thing 4 times now, with a different organizational principle each time. I was 50 pages in to the last version when I decided I should use the regular mapping paradigm (or at least my own version of it), and also that I should write for complete neophytes instead of people with an understanding of the fundamentals of JI. At one point I was also going to leave JI out of it entirely, but I've since found a pretty good middle ground (I think).

> I'd like to add that I'd be more open to all of the output from these
> alternative theories, John's and Michael's and everyone else's, if
> people would start talking more about triads than dyads. Or preferably
> triads with the root being doubled down an octave, which I guess are
> tetrads. I just don't care about dyadic music, and you can take a lot
> of "good" dyads and smush them into a chord that sounds terrible.

Yep. Agreed. However, you can figure out what chords are available in a tuning by looking at the dyads...it's just that it is important to see how various dyads combine into larger harmonic series chunks.

Of course, chord concordance is not very well understood, compared to dyadic concordance. We just don't have all the pieces yet. No theory is as reliable as our ears, but even then the gray areas are vast.

-Igs

On Sun, Mar 20, 2011 at 2:46 AM, cityoftheasleep
<igliashon@...> wrote:
>
> > How upcoming is it?
>
> Well, I've almost finished the first draft of the "Microtonality 101" section, which is the "hard" part. Dealing with the EDOs will be straight-forward enough. Hopefully I won't decide to change horses mid-stream again, because I've re-written the damn thing 4 times now, with a different organizational principle each time. I was 50 pages in to the last version when I decided I should use the regular mapping paradigm (or at least my own version of it), and also that I should write for complete neophytes instead of people with an understanding of the fundamentals of JI. At one point I was also going to leave JI out of it entirely, but I've since found a pretty good middle ground (I think).

Well, let me know when it upcomes. If it's something that the average
trained musician can read, I'll pimp it to all the music school crew
on Facebook. They've been hearing me ramble on about overtones for
about 4 years now, and they still have no idea what I'm talking about
at all.

> > I'd like to add that I'd be more open to all of the output from these
> > alternative theories, John's and Michael's and everyone else's, if
> > people would start talking more about triads than dyads. Or preferably
> > triads with the root being doubled down an octave, which I guess are
> > tetrads. I just don't care about dyadic music, and you can take a lot
> > of "good" dyads and smush them into a chord that sounds terrible.
>
> Yep. Agreed. However, you can figure out what chords are available in a tuning by looking at the dyads...it's just that it is important to see how various dyads combine into larger harmonic series chunks.

16/15 is a fantastic interval. A bunch of stacked 16/15's is not a
fantastic chord.

> Of course, chord concordance is not very well understood, compared to dyadic concordance. We just don't have all the pieces yet. No theory is as reliable as our ears, but even then the gray areas are vast.

If the stuff about minorness being related to tonalness didn't have
your brain melting out of your ears, I won't try to convince you of
anything at all then :)

-Mike

>
> Unfortunately, nothing about the workings of the brain or the inner
> ear are mathematically elegant. This is partly why they're so hard to
> model.
>

I just don't believe that. It depends on what is being modeled and for what
reason. For instance, the Weber-Fechner law is simple, and it describes a
wide range of phenomena, including Bayesian inference with 2 hypotheses.

Many people have come onto these lists and decided that all music can
be explained by secret patterns in numbers. In contrast, other folks
have decided that things like psychoacoustics play more of a role.
Coming up with a mathematically elegant, quick "rule of thumb" that
approximates the behavior of more complex models is a worthy goal. I
don't think John claims that this is what he's doing, but rather that
his equation is a psychoacoustic model in and of itself.

John claimed to have made the "rules" of (Western) music theory redundant, I
believe. I don't know what else he claims, but the classical theory, at
least, as commonly taught, is not a direct psychoacoustic model. I think
John has retracted his prior statement.

What's more important is to know the concepts behind each model.
> Tenney height matches a lot of intervals a little too "lowly," namely
> things like 500000/400001, which are indistinguishable from 5/4. But
> the point of Tenney height is just to reflect that the consonance of
> intervals tends to roughly follow an n*d pattern.

I just wonder if there is a simple way to extend the concept of Tenney
height that is productive - like you say, even as a rule of thumb.

You are allowed to be a fan of John's model if you want. Again, the
> important thing is to understand the concepts at play here.

True, I will definitely look into HE. I didn't find any mention on the
otherwise excellent xenharmonic wiki.

Yes. Perhaps I misspoke. Looks like it's saying that entropy is the
expected value of self-information, and surprisal is self-information.
I was taught that entropy is self-information, which Wikipedia
acknowledges is a common usage of the word here:

"The term self-information is also sometimes used as a synonym of
entropy, i.e. the expected value of self-information in the first
sense, because I(X;X) = H(X), where I(X;X) is the mutual information
of X with itself.[1]"

Right, that was one reason I used the word surprisal instead of
self-information, as well as the fact that it fits in well with musical
notions of predictability.

Here's a somewhat decent page on it:

http://sonic-arts.org/td/erlich/entropy.htm

Note that it doesn't represent critical band effects, but 60 cents
turns out to be pretty dissonant anyways for other reasons.

Thank you, just what I needed!

It's not elegant at all, but it's accurate. The Fourier Transform is
also not elegant, but is a staple of just about everything that we do
here. If you want to come up with an elegant, quick, shorthand way to
figure out how consonant an interval, some parameter averaging
critical band dissonance and ease of F0 estimation is probably a good
way to do it.

I don't think there's any mystery as to why his formula sometimes
"works," except for the holes that have already been mentioned, and I
don't think there should be any difficulty in interpreting it either.

Tenney height < 100 would probably work roughly as well.

The Fourier transform is mathematically very elegant and useful in many
applications, not just audio signal processing, and it's physically elegant
as well, since a simple prism can perform one. It's possible HE has similar
traits. If so, please excuse my ignorance.

> Depends on what is meant by better, but I won't argue that HE models well
> the variables it measures. However, to me, that doesn't imply construct
> validity or mean that it is the most effective way to make music. I would
> have to tinker much more.
>
> All of these criticisms apply a million times more to John's formula
than to harmonic entropy.

Sorry for my nebulous statement there; I'm thinking more in a pragmatic
sense about music-making here. Certainly HE can be incorporated into
instrument design, but is it something that provides intuition in a
performance setting? I understand that a basketball player is not going to
be deriving the range equation during the final seconds of a game, but they
are going to be following certain rules of play. Does HE have an intuitive
set of "rules" that are derivable from first principles?

So I'm going to shut my mouth for a little while on this subject until I've
really looked into HE. I'm certainly glad its out there to explore.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Well, let me know when it upcomes. If it's something that the average
> trained musician can read, I'll pimp it to all the music school crew
> on Facebook. They've been hearing me ramble on about overtones for
> about 4 years now, and they still have no idea what I'm talking about
> at all.

I intend to run the first draft by a few experts, a few teachers, and a few neophytes to see if I did my job right, and then I will probably edit the shit out of it again until it's finally something any jack-ass who's ever tuned a guitar could fathom (if he/she wanted to fathom it, that is)

> 16/15 is a fantastic interval. A bunch of stacked 16/15's is not a
> fantastic chord.

???

> If the stuff about minorness being related to tonalness didn't have
> your brain melting out of your ears, I won't try to convince you of
> anything at all then :)

I still don't agree the minorness is a function of a one-dimensional variable. I do think that chords paired in otonal/utonal relationship (i.e. 4:5:6 vs 1/(4:5:6)) tend to have reciprocal emotional qualities, but we have minor chords that are otonal (6:7:9), and otonal and rooted (16:19:24). But every time I remind you of this, you tell me it doesn't refute anything you're saying and point back to this examples you gave of detuning the 5th harmonic and then adding a bunch of higher harmonics successively to push the chord back toward "major" (which I did not believe were successful in demonstrating anything except that in large chords it is harder to notice a single mis-tuned voice). So clearly I've never understood what you mean about minorness being a function of tonalness. The only way I know how to understand that claim leads to it being refuted by either or both of the example chords I gave.

-Igs

Me>>"I don't bother with 25 or higher EDOs because they contain intervals narrower than 48.18855 cents which I consider to be illegal (see message number 96914 "Crows")."

Igs>>"I'd probably draw the line at 30 rather than 25. I've written some music in 25 where the 48-cent interval is used as a semitone(!) melodically, and it works surprisingly well. At 40 cents though things are definitely breaking down, sounding more like a quaver and less like a step."

Hi Igs, it's nice to see that at least one person agrees with me that there should be a narrowness cut-off point for melodic intervals (i.e. melodic intervals narrower than the cut-off point should be "illegal").

John.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi John,
>
> I did this same task a while ago, except that I included intervals in the 2nd octave as well, and used slightly more tolerance in what I consider "good", such that I would add the following intervals to your list:
> 3/1, 5/2, 7/2, 7/3, 9/4, 10/3, 11/3, 11/4, 11/5, 11/9, 12/5, 13/4, 13/5, 13/6, and 15/4 (I may have forgotten a few). This definitely alters the gamut. Also, I allowed a broader tolerance of up to 8 or 9 cents (nothing too rigorously-defined, as different ratios seem to vary in their sensitivity to mistuning). If you like I can post what I came up with, but it will also be available in my upcoming e-book.
>
> >
> It looks like 23EDO is the most versatile with seven good intervals occurring which is strange because I've rarely seen 23EDO mentioned.
> >
>
> Probably because one can't play common-practice music in 23. I love it, though. It has a lot of good possibilities. It's excellent for music based on 5:6:7 or 3:5:7 harmonic triads. Someday I might get a guitar fretted up to it, though 23's cutting it a bit narrow.
>
> >
> I don't bother with 25 or higher EDOs because they contain intervals narrower than 48.18855 cents which I consider to be illegal (see message number 96914 "Crows").
> >
>
> I'd probably draw the line at 30 rather than 25. I've written some music in 25 where the 48-cent interval is used as a semitone(!) melodically, and it works surprisingly well. At 40 cents though things are definitely breaking down, sounding more like a quaver and less like a step.
>
> -Igs
>

Mike>>"I just don't care about dyadic music, and you can take a lot
of "good" dyads and smush them into a chord that sounds terrible."

I challenge you to find a single "bad" chord an octave or less wide where every dyad in the chord (i.e. every note paired with every other note in the chord) occurs in this list...9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 10/7, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 11/6, 13/7 and 2/1. There are plenty of good chords possible here (an octave or less wide) that have as many as five notes with all the dyads in the chord occurring in the list above.

It seems logical to me to start with dyads first (not triads) and identify which of them are good because in my approach good triads are built upon good dyads.

John.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Mar 19, 2011 at 11:31 PM, cityoftheasleep
> <igliashon@...> wrote:
> >
> > Hi John,
> >
> > I did this same task a while ago, except that I included intervals in the 2nd octave as well, and used slightly more tolerance in what I consider "good", such that I would add the following intervals to your list:
> > 3/1, 5/2, 7/2, 7/3, 9/4, 10/3, 11/3, 11/4, 11/5, 11/9, 12/5, 13/4, 13/5, 13/6, and 15/4 (I may have forgotten a few). This definitely alters the gamut. Also, I allowed a broader tolerance of up to 8 or 9 cents (nothing too rigorously-defined, as different ratios seem to vary in their sensitivity to mistuning). If you like I can post what I came up with, but it will also be available in my upcoming e-book.
>
> How upcoming is it?
>
> I'd like to add that I'd be more open to all of the output from these
> alternative theories, John's and Michael's and everyone else's, if
> people would start talking more about triads than dyads. Or preferably
> triads with the root being doubled down an octave, which I guess are
> tetrads. I just don't care about dyadic music, and you can take a lot
> of "good" dyads and smush them into a chord that sounds terrible.
>
> -Mike
>

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:

> Really I just like the look and feel of John's formula, and I tend to trust
> mathematical elegance.

Then you should prefer Tenney height, which uses a simpler formula and one which has elegant, mathematically useful consequences.

I think such a formula would be a useful tool, and
> that it could even reveal a pragmatic psychophysical truth. I am just
> starting down my microtonal study, but I wonder if Tenney height might rate
> certain (few) intervals a little to highly.

It could, but when I did a side-by side comparison I thought it came out a little better than John's formula.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Mar 19, 2011 at 11:53 PM, Daniel Nielsen <nielsed@...> wrote:
> >
> > Although you may not consider it valid for 3+-ads, John does at least explain his working theory for those chords in his book, which fits into the same framework (and explains some of what was for me confusing terminology about independent note "strength").
>
> As I've said before, I think that John's ideas are somewhat on the
> right track. His equation is, roughly, a weighted average of critical
> band roughness and Tenney Height, which are both types of dissonance.

I looked up Tenney Height on Tonalsoft and it seems to be x multiplied by y (x/y is a JI interval). x*y does not occur in my formula...

(2 + 1/x + 1/y - diss(x,y))/2

diss(x,y) is a function.

If y/x <= 0.9375, diss = y/x

If y/x > 0.9375, diss = (1 - y/x)*15

>
> I don't like the arbitrary 256/255 cutoff, and I don't like that he's

As I've explained before the 256/255 was an educated guess backed up with listening tests, not exactly arbitrary. 6.776 cents (256/255) tempering sounds acceptable to my ear and a slightly wider tempering (8 cents) sounds unacceptable.

> just picked one way to average these and decided that it's "good," and
> I don't like that he's using Tenney Height instead of HE for the

Again, I'm not using Tenney Height.

> concordance part of the equation. I also don't like that not a lot of
> explanatory power has been given here and that it's presented as a
> simple "master equation," but other than that it isn't too bad.

I've explained my formula and Interval Evaluation Calculator program countless times on this list, you probably never bothered to read these posts since I "must be wrong".

>
> But you still can't just take dyads together and smush them randomly
> together into chords and write consonant music from it.

I disagree.

John.

>
> -Mike
>

MikeB>"16/15 is a fantastic interval. A bunch of stacked 16/15's is not a fantastic chord."

   I repeatedly (12 times in a row) tested alternating (in random order) between 16/15 and 12/11 to blind test my girlfriend.  Every single time without exception she prefered the 12/11.  When stacking two 16/15's vs. two 12/11 I got the same result...12/11 won every time, no exceptions, no hesitations as to answering which sounds better.

   Not to say this will hold for every listener but...it seems clear to me once you duck under about 12/11...critical band dissonance between the root tones begins to take over as the most unbearable factor in a single dyad and most certainly stacked dyads (not that the overtone-alignment doesn't matter, but that it matters much less than usual).

  Granted though, I have a hunch 16/15 is infinitely more desirable than something like 41/40 (the two tones played at once)...where your mind often can't figure out "is this two notes or not?"

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> Hi Igs, it's nice to see that at least one person agrees with me that there should be a narrowness cut-off point for melodic intervals (i.e. melodic intervals narrower than the cut-off point should be "illegal").

I've experimented with music where all of the steps are "illegal". It's an interesting effect, the challenge being to get harmony also. The melody line oozes and oils around.

Gene,

my harmony interval formula

(2 + 1/x + 1/y - diss(x,y))/2

applies to sine wave tones only and not to complex tones. I wrote a program that uses the formula above to work out the values of harmony intervals with complex tones and 'regular' timbres.

Want to try that comparison again using sine wave tones?

Or were you referring to my *melody* formula?

2/x + 2/y

This should be good for both sine wave tones and complex tones with a 'regular' timbre.

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@> wrote:
>
> > Really I just like the look and feel of John's formula, and I tend to trust
> > mathematical elegance.
>
> Then you should prefer Tenney height, which uses a simpler formula and one which has elegant, mathematically useful consequences.
>
> I think such a formula would be a useful tool, and
> > that it could even reveal a pragmatic psychophysical truth. I am just
> > starting down my microtonal study, but I wonder if Tenney height might rate
> > certain (few) intervals a little to highly.
>
> It could, but when I did a side-by side comparison I thought it came out a little better than John's formula.
>

On Sun, Mar 20, 2011 at 12:59 PM, cityoftheasleep
<igliashon@...> wrote:
>
> > If the stuff about minorness being related to tonalness didn't have
> > your brain melting out of your ears, I won't try to convince you of
> > anything at all then :)
>
> I still don't agree the minorness is a function of a one-dimensional variable. I do think that chords paired in otonal/utonal relationship (i.e. 4:5:6 vs 1/(4:5:6)) tend to have reciprocal emotional qualities, but we have minor chords that are otonal (6:7:9), and otonal and rooted (16:19:24).

6:7:9 and 16:19:24 do sound "minor." But you have to keep in mind that
there's not just one f0 to be found in the chord, but quite a lot. (f0
is psychoacousticalese for "fundamental frequency," and I'm going to
use that term here because it's a lot easier to type.)

In the case of a major chord, 4:5:6, the whole thing points to an f0
at 1, but individual f0's can be found at... 4, 5, and 6. This is why
you can still hear the notes in the chord and not just the f0 itself.

In the case of 6:7:9, you have a very strong f0 at 3, which is the GCD
of 6 and 9. Then you have a weaker one at 1, and then you have f0's at
each individual note. So if you hear that 6:7:9 as a "resolved" chord
(which is another term borrowed from the psychoacoustics literature),
meaning as a solid unit that points to 1, then it will sound "otonal,"
meaning that it will also sound like there's a 1 popping out that is
not octave-equivalent to the 6. On the other hand, if you don't hear
it that way, but you hear the 6 as the root, meaning you're hearing
the 6:9 as the main f0 here with the 7 as some weakly related crap,
then it -will- sound "minor."

If you hear the 16:19:24 as pointing to a 1 way down at the bottom of
the earth, it will sound weakly "major." If you hear it as pointing to
a 1 higher up, such that the 16:24 is heard as just a 2:3 and the 19
is heard as weakly related or maybe even inharmonic, it will sound
"minor."

Check out this thread for some interesting JI chords that I think
sound "minor": /tuning/topicId_96260.html#96260

> But every time I remind you of this, you tell me it doesn't refute anything you're saying and point back to this examples you gave of detuning the 5th harmonic and then adding a bunch of higher harmonics successively to push the chord back toward "major" (which I did not believe were successful in demonstrating anything except that in large chords it is harder to notice a single mis-tuned voice).

That was what I said a long time ago. It still has some perhaps
indirect application but let's get past the mistuning, or "depriming"
or whatever I was saying back then, because that's not what I think
anymore.

-Mike

>
> > Really I just like the look and feel of John's formula, and I tend to
> trust
> > mathematical elegance.
>
> Then you should prefer Tenney height, which uses a simpler formula and one
> which has elegant, mathematically useful consequences.
>
>
I don't know where to go with Tenney height. To a number theorist, maybe
there's the world in the multiplication of the numbers in a reduced
fraction. What I see in John's work is a form that looks very much like many
other forms in simple Bayesian and physical systems. John's formula is also
easily written using Tenney height as a factor. He seems to have made
careful aesthetic judgments during his work, although I cannot attest to
their validity or explanation. He also almost accidentally, it seems,
classified dyads by their surprisal scores.

John's system makes several assumptions, which he identifies clearly, and
which may be modified to improve it.

>
> >I think such a formula would be a useful tool, and
> > that it could even reveal a pragmatic psychophysical truth. I am just
> > starting down my microtonal study, but I wonder if Tenney height might
> rate
> > certain (few) intervals a little to highly.
>
> It could, but when I did a side-by side comparison I thought it came out a
> little better than John's formula.
>

Gene, I'd like to tinker with John's formula a little to see if it could be
modified. Michael seemed to agree with most of its results more or less. I'd
like to identify key Tenney-height-sorted intervals that might be improved.
My guesses may be off base, though. (Of course, there is also the question
of the extent of a true "deep structure" in music.)

On Sun, Mar 20, 2011 at 10:42 AM, Daniel Nielsen <nielsed@...> wrote:
>>
>> Unfortunately, nothing about the workings of the brain or the inner
>> ear are mathematically elegant. This is partly why they're so hard to
>> model.
>
> I just don't believe that. It depends on what is being modeled and for what reason. For instance, the Weber-Fechner law is simple, and it describes a wide range of phenomena, including Bayesian inference with 2 hypotheses.

Then I don't understand what exactly you find "elegant," nor why
elegance should be a precursor to validity.

>> Yes. Perhaps I misspoke. Looks like it's saying that entropy is the
>> expected value of self-information, and surprisal is self-information.
>> I was taught that entropy is self-information, which Wikipedia
>> acknowledges is a common usage of the word here:
>>
>> "The term self-information is also sometimes used as a synonym of
>> entropy, i.e. the expected value of self-information in the first
>> sense, because I(X;X) = H(X), where I(X;X) is the mutual information
>> of X with itself.[1]"
>
> Right, that was one reason I used the word surprisal instead of self-information, as well as the fact that it fits in well with musical notions of predictability.

I don't see in what sense you claim that John's formula has anything
to do with surprisal then. Would musical notions of predictability not
have more to do with prior exposure?

>> I don't think there's any mystery as to why his formula sometimes
>> "works," except for the holes that have already been mentioned, and I
>> don't think there should be any difficulty in interpreting it either.
>>
>> Tenney height < 100 would probably work roughly as well.
>
> The Fourier transform is mathematically very elegant and useful in many applications, not just audio signal processing, and it's physically elegant as well, since a simple prism can perform one. It's possible HE has similar traits. If so, please excuse my ignorance.

If you think that the Fourier transform is mathematically very
elegant... then there are lots of things that are mathematically
elegant. The math behind that is too much for most folks here. But
again, John has not shown how his model is an elegant approximation to
a more complex behavior. He came up with a 256/255 (why this
particular number?) field of attraction for each interval and set some
other arbitrary cutoffs that he thought were "good."

11/7 and 8/5 both appear as "good" intervals - although they can
perhaps be distinguished in practice, they both sound very "similar."
The fact that they sound similar is not an immutable necessity of the
universe and can be explained by the way that the ear/brain system is
set up. HE, furthermore, explains this. John's formula does not.

>> > Depends on what is meant by better, but I won't argue that HE models well the variables it measures. However, to me, that doesn't imply construct validity or mean that it is the most effective way to make music. I would have to tinker much more.
>>
>> All of these criticisms apply a million times more to John's formula
>> than to harmonic entropy.
>
> Sorry for my nebulous statement there; I'm thinking more in a pragmatic sense about music-making here. Certainly HE can be incorporated into instrument design, but is it something that provides intuition in a performance setting? I understand that a basketball player is not going to be deriving the range equation during the final seconds of a game, but they are going to be following certain rules of play. Does HE have an intuitive set of "rules" that are derivable from first principles?

No. I think that's better than having an intuitive set of "rules" that
is wrong. Sue me.

-Mike

On Sun, Mar 20, 2011 at 1:38 PM, john777music <jfos777@...> wrote:
>
> Mike>>"I just don't care about dyadic music, and you can take a lot
>
> of "good" dyads and smush them into a chord that sounds terrible."
>
> I challenge you to find a single "bad" chord an octave or less wide where every dyad in the chord (i.e. every note paired with every other note in the chord) occurs in this list...9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 10/7, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 11/6, 13/7 and 2/1. There are plenty of good chords possible here (an octave or less wide) that have as many as five notes with all the dyads in the chord occurring in the list above.

840:1155:1320:1540:1848 is pretty dark, but I guess usable. But if
that's how you're doing things above then you're doing something very
close to odd limit, which is probably a good way to build triads.
Nonetheless, just saying which dyads are in a scale doesn't say
anything about how the scale permits you to actually form them into
triads that meet your criterion above. You should throw that
information out there.

-Mike

On Sun, Mar 20, 2011 at 2:02 PM, john777music <jfos777@...> wrote:
>
> I looked up Tenney Height on Tonalsoft and it seems to be x multiplied by y (x/y is a JI interval). x*y does not occur in my formula...
>
> (2 + 1/x + 1/y - diss(x,y))/2

1/x + 1/y = (x+y)/(x*y). x*y is Tenney Height, and x+y is something
that wasn't used much here but I guess could be called "Mann height"
from Paul's experiments with it while he was designing HE. So your
formula has a 1/(Tenney Height) term, meaning that more complex
intervals get penalized.

For large x and y, x*y >> x+y.

The diss(x,y) is basically a function that penalizes the interval once
it moves too closely within a critical bandwidth. Intervals with too
much roughness get penalized.

Your formula is a weighted average of something that, for all intents
and purposes, is an average of the inverse of its complexity (in this
case measured by something asymptotically approaching Tenney Height)
and critical band roughness.

> > I don't like the arbitrary 256/255 cutoff, and I don't like that he's
>
> As I've explained before the 256/255 was an educated guess backed up with listening tests, not exactly arbitrary. 6.776 cents (256/255) tempering sounds acceptable to my ear and a slightly wider tempering (8 cents) sounds unacceptable.

How is it educated? Why the random numbers 256 and 255, divided with
one another?

Tempering 5/4 sharp by 14 cents sounds acceptable to most of the
western world, as does tempering 6/5 16 cents flat, so I'm not sure
your results apply.

> > concordance part of the equation. I also don't like that not a lot of
> > explanatory power has been given here and that it's presented as a
> > simple "master equation," but other than that it isn't too bad.
>
> I've explained my formula and Interval Evaluation Calculator program countless times on this list, you probably never bothered to read these posts since I "must be wrong".

The only explanation that I have read is that this is the formula
you've picked because you like the results. I am actually trying to
help your case here by offering a psychoacoustic angle that might
validate, conceptually, what you're doing. If your goal is just to
come up with some simple rules of thumb that might help someone find
useful intervals, then that's fine. But as far as I know, you have not
given an explanation about why these are supposed to be the magic
numbers, and you have not addressed to my satisfaction what the reason
is supposed to be.

There's also the fact that you haven't explained why a 12-tet major
chord sounds great to almost every single person living on the planet,
but contains no intervals that you'd mark "good."

-Mike

On Sun, Mar 20, 2011 at 2:08 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"16/15 is a fantastic interval. A bunch of stacked 16/15's is not a fantastic chord."
>
>    I repeatedly (12 times in a row) tested alternating (in random order) between 16/15 and 12/11 to blind test my girlfriend.  Every single time without exception she prefered the 12/11.  When stacking two 16/15's vs. two 12/11 I got the same result...12/11 won every time, no exceptions, no hesitations as to answering which sounds better.

I don't care. 16/15 is used in chords all the time. OK, 12/11 is less
harsh for critical bandwidth reasons. So what? To say that 16/15 is
unusable as a harmonic interval is ridiculous. As an obvious example,
and one I keep making and making and making again, there are millions
of songs written that utilize major 7 chords in inversion. We should
take care when devising various microtonal "theories" around here that
we aren't breaking away from reality.

>    Not to say this will hold for every listener but...it seems clear to me once you duck under about 12/11...critical band dissonance between the root tones begins to take over as the most unbearable factor in a single dyad and most certainly stacked dyads (not that the overtone-alignment doesn't matter, but that it matters much less than usual).

For you. But I've seen lots of evidence to think that you can train
yourself to identify tones from within a beating structure, up to a
point.

-Mike

> and x+y is something
> that wasn't used much here but I guess could be called "Mann height"
> from Paul's experiments with it while he was designing HE.
>

That's catchy, Manhattan (distance)->Mann-heightened.

On Sun, Mar 20, 2011 at 5:19 PM, Daniel Nielsen <nielsed@...> wrote:
>
>
>>
>> and x+y is something
>> that wasn't used much here but I guess could be called "Mann height"
>> from Paul's experiments with it while he was designing HE.
>
> That's catchy, Manhattan (distance)->Mann-heightened.

Indeed. Maybe that's why he did it, although Tenney height is also a
type of L1 norm, so I dunno...

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Mar 20, 2011 at 1:38 PM, john777music <jfos777@...> wrote:
> >
> > Mike>>"I just don't care about dyadic music, and you can take a lot
> >
> > of "good" dyads and smush them into a chord that sounds terrible."
> >
> > I challenge you to find a single "bad" chord an octave or less wide where every dyad in the chord (i.e. every note paired with every other note in the chord) occurs in this list...9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 10/7, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 11/6, 13/7 and 2/1. There are plenty of good chords possible here (an octave or less wide) that have as many as five notes with all the dyads in the chord occurring in the list above.
>
> 840:1155:1320:1540:1848 is pretty dark, but I guess usable.

There you go, if all the dyads in a chord are "usable" then the chord should be "usable". I've identified every single possible chord (ranging from 2 notes to 5 notes, an octave or less wide where all the dyads are good within 6.776 cents accuracy) that occurs in my Blue Temperament scale and they *all* sound "usable" without exception.

But if
> that's how you're doing things above then you're doing something very
> close to odd limit, which is probably a good way to build triads.
> Nonetheless, just saying which dyads are in a scale doesn't say
> anything about how the scale permits you to actually form them into
> triads that meet your criterion above. You should throw that
> information out there.

I'm pretty sure I've already mentioned on the list that if all of the dyads in a chord are good then the chord should be good. Identify the good dyads and then use them to build good triads. For me if a chord has a single bad dyad then the chord is no good which you will disagree with because you use 16/15 in some of your jazz chords which for me is illegal.

John.

>
> -Mike
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> 6:7:9 and 16:19:24 do sound "minor." But you have to keep in mind that
> there's not just one f0 to be found in the chord, but quite a lot. (f0
> is psychoacousticalese for "fundamental frequency," and I'm going to
> use that term here because it's a lot easier to type.)
>
> In the case of a major chord, 4:5:6, the whole thing points to an f0
> at 1, but individual f0's can be found at... 4, 5, and 6. This is why
> you can still hear the notes in the chord and not just the f0 itself.

Okay, but then shouldn't 2:3:4 sound "even more major" than 4:5:6, because it is closer to the f0 at 1? And shouldn't 2:6:10 sound less major than 4:5:6? Also, I'm sure I've said this before, but what about 5:6:7? I can't think of a single emotional quality I would ascribe to both that chord and 4:5:6. 5:6:7 is a brash, strident, even *angry* chord. Almost night-and-day emotionally-speaking compared to 4:5:6. And yet 6:7:8, 7:8:9, and 8:9:10 all sound *less* strident and brash, feeling more "floaty" and "soft"...and in fact those three triads feel fairly similar to one another to my ears. I wouldn't call any of them major or minor, either; they all sound suspended to me. And, hey, what about 6:8:9? That one *doesn't* sound minor, but by your method of analysis, it's not a whole lot different from 6:7:9. Or how about 9:10:12? Shouldn't that sound minor too?

All I can say is that to me, the only chords that sound "minor" are chords that sound something like minor triads, which is to say any inversion of a chord of 0 cents, 260-340 cents, and 660-740 cents. Adding a fourth interval or tuning any of those intervals out of the ranges I gave always leads (at least to my ears) to a different emotional place. I don't think any of the chords you've suggested that don't fit this pattern share any emotionally-significant qualities with the chords that do. I can't really buy your theory unless you can produce a triad outside of the ranges I specified that makes me think of/feel something remarkably similar to a triad within the ranges I specified.

-Igs

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:

> I don't know where to go with Tenney height. To a number theorist, maybe
> there's the world in the multiplication of the numbers in a reduced
> fraction. What I see in John's work is a form that looks very much like many
> other forms in simple Bayesian and physical systems.

The key advantage of Tenney is that it is diagonizable, by which I mean that 3/2 has the same height as 6, etc. It is, in essence, the L1 norm with weighted primes as the basis, and if you are willing to use the L2 norm instead, you obtain further mathematical advantages.

John's formula is also
> easily written using Tenney height as a factor. He seems to have made
> careful aesthetic judgments during his work, although I cannot attest to
> their validity or explanation.

He has his views, and other people have theirs, and his personal taste is not a universal. To my ears odd limits are just as good and have some practical advantages, as well as being less subjective.

> John's system makes several assumptions, which he identifies clearly, and
> which may be modified to improve it.

It seems pretty ad hoc to me.

> Gene, I'd like to tinker with John's formula a little to see if it could be
> modified. Michael seemed to agree with most of its results more or less. I'd
> like to identify key Tenney-height-sorted intervals that might be improved.

OK. TH rates 7/6 as better than 9/5, 9/7 as better than 11/6, 11/8 as better than 13/7, 11/10 as better than 15/8, 13/11 as better than 16/9, 15/13 as better than 18/11 and 16/13 as better than 20/11. From the point of view of Tenney height, O'Sullivan's measure is giving a small bonus to having smaller denominators and larger numerators, and so tends to give the nod to wider intervals.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> I'm pretty sure I've already mentioned on the list that if all of the dyads in a chord are good > then the chord should be good. Identify the good dyads and then use them to build good
> triads. For me if a chord has a single bad dyad then the chord is no good which you will
> disagree with because you use 16/15 in some of your jazz chords which for me is illegal.

I think Mike's point is that you can have a scale where the dyads all look good with the root but are not good with each other. It's easy enough to construct a scale like this in JI. Here's one off the top of my head:

1/1-8/7-9/8-6/5-9/7-11/8-10/7-3/2-11/7-12/7-7/4-13/7-2/1

There are not very many good triads in this scale.

So you have to also account for how the dyads relate to each other, which (as it turns out) is pretty much identical to considering triads.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Mar 20, 2011 at 2:02 PM, john777music <jfos777@...> wrote:
> >
> > I looked up Tenney Height on Tonalsoft and it seems to be x multiplied by y (x/y is a JI interval). x*y does not occur in my formula...
> >
> > (2 + 1/x + 1/y - diss(x,y))/2
>
> 1/x + 1/y = (x+y)/(x*y). x*y is Tenney Height,

Yeah, but what about the 1/x + 1/y? This changes things a lot so it's not *purely* about Tenney Height and the occurrence of (x*y) above is a coincidence. In my book I state that I tested 1/(x*y) and found one clear inconsistency and so ruled it out.

and x+y is something
> that wasn't used much here but I guess could be called "Mann height"
> from Paul's experiments with it while he was designing HE. So your
> formula has a 1/(Tenney Height) term, meaning that more complex
> intervals get penalized.
>
> For large x and y, x*y >> x+y.
>
> The diss(x,y) is basically a function that penalizes the interval once
> it moves too closely within a critical bandwidth. Intervals with too
> much roughness get penalized.
>
> Your formula is a weighted average of something that, for all intents
> and purposes, is an average of the inverse of its complexity (in this
> case measured by something asymptotically approaching Tenney Height)
> and critical band roughness.
>
> > > I don't like the arbitrary 256/255 cutoff, and I don't like that he's
> >
> > As I've explained before the 256/255 was an educated guess backed up with listening tests, not exactly arbitrary. 6.776 cents (256/255) tempering sounds acceptable to my ear and a slightly wider tempering (8 cents) sounds unacceptable.
>
> How is it educated? Why the random numbers 256 and 255, divided with
> one another?

Here's the "guess". Powers of 2 kept on appearing in my research. 1/1 to 2/1 seems like the best range for a melody, both notes sounding more resolved than any other note in between or beyond.

Testing dissonance using sine wave tones it seemed that either 16/15 or 17/16 was the most dissonant sine wave interval, 16 is a power of 2.

In melody (my formula is 2/x + 2/y for both sine waves and complex tones with a 'regular' timbre) it seems that any melodic interval with a value between 2.0 and 4.0 sounds like a "Super Major". Any value between 1.0 and 1.9999 sounds like an "ordinary" Major (I call it a Blue Major). Any value between 0.5 and 0.9999 sounds like an "ordinary" minor (I call it a Blue Minor). Any value between 0.25 and 0.4999 sounds like an Ultra Minor. All of the above sound "sweet" to me. Next, a melodic interval with a value between 0.125 and 0.2499 sounds "tolerable" to me but not sweet. Finally an interval with a value of less than 0.125 sounds to me to be "intolerable". Each successive range is half of the previous range...1, 1/2, 1/4, 1/8, 1/16, 1/32.

So I guessed at 256/255 (6.776 cents) because 256 is power of 2 and a few listening tests confirmed (to me at least) that it was an acceptable tempering value. Next I tried an 8 cent tempering and could just about detect a slight hint of dissonance (I'm pretty fussy) and so ruled it out. With dissonance I chose 16/15 (15 is one less than 16). Similarly I chose 256/255 because 255 is one less than 256. Think of 1/16 or 1/256.

>
> Tempering 5/4 sharp by 14 cents sounds acceptable to most of the
> western world, as does tempering 6/5 16 cents flat, so I'm not sure
> your results apply.

I worked out my formulas for *me* and not the western world. Sure you can "get away" with the mistunings you list above but I'm aiming for perfection or close enough to it.

>
> > > concordance part of the equation. I also don't like that not a lot of
> > > explanatory power has been given here and that it's presented as a
> > > simple "master equation," but other than that it isn't too bad.
> >
> > I've explained my formula and Interval Evaluation Calculator program countless times on this list, you probably never bothered to read these posts since I "must be wrong".
>
> The only explanation that I have read is that this is the formula
> you've picked because you like the results.

This is partly true, the results do look "cute", magic numbers if you will. I guessed at hundreds of possible combinations of formulas and slowly eliminated them one by one (over a 14 year period making lots of mistakes) until I arrived at what I considered to be the most likely formula. It's not so much that "it must be formula 'a'" but rather "it can't be b or c or d or e or f or g or h etc".

I am actually trying to
> help your case here by offering a psychoacoustic angle that might
> validate, conceptually, what you're doing. If your goal is just to
> come up with some simple rules of thumb that might help someone find
> useful intervals, then that's fine. But as far as I know, you have not
> given an explanation about why these are supposed to be the magic
> numbers, and you have not addressed to my satisfaction what the reason
> is supposed to be.
>
> There's also the fact that you haven't explained why a 12-tet major
> chord sounds great to almost every single person living on the planet,
> but contains no intervals that you'd mark "good."

I never said that. In the 12TET open E Major chord on a guitar (E,B,E,G#,B,E) all the intervals that occur are within 6.776 cents of any of my own "good" intervals except the intervals that contain the G#. And why should it be my responsibility to explain this anyway?

John.

>
> -Mike
>

John>"Testing dissonance using sine wave tones it seemed that either 16/15 or
17/16 was the most dissonant sine wave interval, 16 is a power of 2."

    Sounds about right.  Sethares says that around middle c, such is the case...although as you get higher in frequency, the critical band gets smaller.  You can't beat the fact the critical band is "curved" in designing scales anyhow...unless you don't use a period (like the octave)...and I've heard many people argue that 7TET is about the best you can get, on average (across all frequencies generally used for root tones in music) for a 7-note scale using a 2/1 period.

>"Next I tried an 8 cent tempering and could just about detect a slight
hint of dissonance (I'm pretty fussy) and so ruled it out."

   Sounds about right, or at least pretty conservative.
  Virtually no one seems to agree on an exact value of what's "too far out" far as tempering.  I've heard anything from 3 cents maximum to about 10 cents maximum depending on the person and (of course) even arguments of 12TET's 14-cent errors being "ok" from 12TET listeners . Plus it seems most people, even the ear-trained bunch on this list, seem to say 7 cents is already quite good.  So 6.776 cents sounds plenty critical/exact enough to me...I even allow 8...certainly a note only 6.776 cents off isn't going to shout itself out as obviously out of tune...

John>"Testing dissonance using sine wave tones it seemed that either 16/15 or
17/16 was the most dissonant sine wave interval, 16 is a power of 2."

    Sounds about right.  Sethares says that around middle c, such is the case...although as you get higher in frequency, the critical band gets smaller.  You can't beat the fact the critical band is "curved" in designing scales anyhow...unless you don't use a period (like the octave)...and I've heard many people argue that 7TET is about the best you can get, on average (across all frequencies generally used for root tones in music) for a 7-note scale using a 2/1 period.

>"Next I tried an 8 cent tempering and could just about detect a slight
hint of dissonance (I'm pretty fussy) and so ruled it out."

   Sounds about right, or at least pretty conservative.
  Virtually no one seems to agree on an exact value of what's "too far out" far as tempering.  I've heard anything from 3 cents maximum to about 10 cents maximum depending on the person and (of course) even arguments of 12TET's 14-cent errors being "ok" from 12TET listeners . Plus it seems most people, even the ear-trained bunch on this list, seem to say 7 cents is already quite good.  So 6.776 cents sounds plenty critical/exact enough to me...I even allow 8...certainly a note only 6.776 cents off isn't going to shout itself out as obviously out of tune...

On Sun, Mar 20, 2011 at 6:06 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > 6:7:9 and 16:19:24 do sound "minor." But you have to keep in mind that
> > there's not just one f0 to be found in the chord, but quite a lot. (f0
> > is psychoacousticalese for "fundamental frequency," and I'm going to
> > use that term here because it's a lot easier to type.)
> >
> > In the case of a major chord, 4:5:6, the whole thing points to an f0
> > at 1, but individual f0's can be found at... 4, 5, and 6. This is why
> > you can still hear the notes in the chord and not just the f0 itself.
>
> Okay, but then shouldn't 2:3:4 sound "even more major" than 4:5:6, because it is closer to the f0 at 1?

There are a lot of things that go into the very loaded word "major."
2:3:4 sounds more "resolved" and "fused" than 4:5:6. As a sonority
starts sounding less resolved, it starts sounding more and more
irritating, scary, terrifying, discordant, whatever, passing through
"minor" on its way there.

> And shouldn't 2:6:10 sound less major than 4:5:6?

2:6:10 is 1:3:5, so no...

> Also, I'm sure I've said this before, but what about 5:6:7? I can't think of a single emotional quality I would ascribe to both that chord and 4:5:6. 5:6:7 is a brash, strident, even *angry* chord. Almost night-and-day emotionally-speaking compared to 4:5:6.

You can "force" 5:6:7 into an unresolved perception by doubling the 5
down an octave. On the other hand, you can force it into a "resolved"
perception by either playing or imagining a 1 such that this becomes
the upper partials of something octave-equivalent to 4:5:6:7.

> And yet 6:7:8, 7:8:9, and 8:9:10 all sound *less* strident and brash, feeling more "floaty" and "soft"...and in fact those three triads feel fairly similar to one another to my ears. I wouldn't call any of them major or minor, either; they all sound suspended to me.

These are good points, but don't have much to do with what I was
getting at, which are the perception of chords that you construct by
deliberately creating different f0's within the chord. Again, see the
semitonal JI thread.

To answer your points, I would say that one way to force an unresolved
perception for most of these chords is to just double the root down an
octave. If that doesn't work, because the resultant chord is still too
tonal, then double it down an octave and also add a 3/2 above the
root, such that you're creating a 1:2:3 pointing to a different f0
than the actual chord is. Make sure you do these all with sine waves.

6:7:8 refers my ears very strongly to 1. If you double the 6 down an
octave, you get 3:6:7:8, which sounds a bit more "minor" than just
6:7:8 by itself. It still sounds like it might be pointing to 1,
however, because of the strength of the 4/3 on the outside. Adding a
3/2 above the 3 gets you 6:9:12:14:16, which sounds even more minor.

I'll skip to 8:9:10 - 8:9:10 resolves to 1 and there's not much wrong
with this. Double the 8 down an octave and it sounds the same. Put
some other notes on the bottom and it may not sound the same. Since
this is a rooted triad, it's not a case where you can create
uncorrelated periodicity information by just doubling the root down an
octave. The fact that you -can- do this for other chords is what is
significant.

7:8:9 to my ears sounds ambiguous and also sounds like while it can be
very clearly heard as 7:8:9, it's on the brink of also sounding like a
sharp 8:9:10. Double the 7 down an octave to get 6:14:16:18 and if
you're using sines, this should just sound like a sharp 4:8:9:10. Do
it with harmonic timbres and things start to sound a bit more
interesting, although not quite minor.

> And, hey, what about 6:8:9? That one *doesn't* sound minor, but by your method of analysis, it's not a whole lot different from 6:7:9. Or how about 9:10:12? Shouldn't that sound minor too?

This is the one interesting exception I've found to the rule and I'm
not sure exactly how to explain it. In this case, doubling the octave
down gives 3:6:8:9, but this doesn't cause the characteristic "minor"
feel. In this case, I think the dominant structures are the 1:2:3 on
the outside and the 6:8 as a 3:4 pointing to 2; I'm not sure if the
8:9 is getting any action or the 6:8:9 as a triad. All I can say to
this is that 4/3 is a very strong interval, and it may just be harder
for the brain to snap it into some kind of "aperiodic" perception than
it is for the 7 in 6:7:9. I haven't found any other exceptions than
this one and perhaps some inversions of it.

> All I can say is that to me, the only chords that sound "minor" are chords that sound something like minor triads, which is to say any inversion of a chord of 0 cents, 260-340 cents, and 660-740 cents. Adding a fourth interval or tuning any of those intervals out of the ranges I gave always leads (at least to my ears) to a different emotional place. I don't think any of the chords you've suggested that don't fit this pattern share any emotionally-significant qualities with the chords that do.

1) Check the semitonal JI thread.
2) It's not that I'm saying that random chords that aren't around
0-300-700 cents will sound, in every sense of the word, "minor."
3) Part of the perception of minorness is a vague kind of polytonality
in which the brain hears multiple harmonic structures that fit
together the wrong way.
4) This part of the perception of minorness can be replicated to
varying degrees in other chords that will obviously not be in the same
triadic range as "minor." Sometimes they don't sound "minor" but
rather "neutral" or sometimes they sound like something completely
different. But they do not sound resolved and tonal and major, which
is the point.

Here's a question - which sounds more like an "even more minor than
minor" version of of the 5-limit minor chord to you? 70:84:105:120,
the 7-limit minor chord, or 40:48:60:75, which places an additional
5/4 above the 3/2? How about 10:12:15:19?

How about these two:

2310:3465:4620:5544:6930:7920:10080:12320 (the 11-limit minor chord),
or 10:15:20:24:30:38:44:52

> I can't really buy your theory unless you can produce a triad outside of the ranges I specified that makes me think of/feel something remarkably similar to a triad within the ranges I specified.

This isn't "my" theory. It's "a" theory. I think I should be able to
discuss how psychoacoustic factors play a role in musical perception
without it turning into some kind of "ownership" thing. And for the
record, Carl is the first person who I'd seen suggest it, as before
this I had taken up Paul's ideas about minorness being caused by
scalar structure.

As for producing a triad, I dunno, but maybe 10:15:16:18:20 will do
the trick. Or perhaps 10:15:16:18:22 or 10:15:16:18:23. I guess
10:15:16 will do as a triad.

A note: Check the semitonal JI thread I linked you to and perhaps
you'll see what I'm getting at. If not, and if you insist on
continually framing this as you have in the past as me trying to
promote "my" theory, instead of two intelligent people having a
discussion about psychoacoustics in music, then I don't really want to
talk about it anymore. I do not claim to have the perfect answer - in
fact, I claim the opposite - but I do not want to discuss stuff like
this anymore unless it's on fair-minded terms.

-Mike

On Sun, Mar 20, 2011 at 6:52 PM, john777music <jfos777@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Sun, Mar 20, 2011 at 2:02 PM, john777music <jfos777@...> wrote:
> > >
> > > I looked up Tenney Height on Tonalsoft and it seems to be x multiplied by y (x/y is a JI interval). x*y does not occur in my formula...
> > >
> > > (2 + 1/x + 1/y - diss(x,y))/2
> >
> > 1/x + 1/y = (x+y)/(x*y). x*y is Tenney Height,
>
> Yeah, but what about the 1/x + 1/y? This changes things a lot so it's not *purely* about Tenney Height and the occurrence of (x*y) above is a coincidence.

It's not a coincidence at all. 1/x + 1/y = (x+y)/(x*y). For large x
and y, this will asymptotically approach Tenney Height. Being as the
whole point of this is to prune the lattice for large x and y, the two
will yield very close results.

> In my book I state that I tested 1/(x*y) and found one clear inconsistency and so ruled it out.

And what is that inconsistency?

> > How is it educated? Why the random numbers 256 and 255, divided with
> > one another?
>
> Here's the "guess". Powers of 2 kept on appearing in my research. 1/1 to 2/1 seems like the best range for a melody, both notes sounding more resolved than any other note in between or beyond.
>
> Testing dissonance using sine wave tones it seemed that either 16/15 or 17/16 was the most dissonant sine wave interval, 16 is a power of 2.

OK, but this is my point. There is no clear explanation for why. Your
explanations are always like "it just seemed to me at the time to be
right."

> In melody (my formula is 2/x + 2/y for both sine waves and complex tones with a 'regular' timbre) it seems that any melodic interval with a value between 2.0 and 4.0 sounds like a "Super Major". Any value between 1.0 and 1.9999 sounds like an "ordinary" Major (I call it a Blue Major). Any value between 0.5 and 0.9999 sounds like an "ordinary" minor (I call it a Blue Minor). Any value between 0.25 and 0.4999 sounds like an Ultra Minor. All of the above sound "sweet" to me. Next, a melodic interval with a value between 0.125 and 0.2499 sounds "tolerable" to me but not sweet. Finally an interval with a value of less than 0.125 sounds to me to be "intolerable". Each successive range is half of the previous range...1, 1/2, 1/4, 1/8, 1/16, 1/32.

I don't see how, mathematically, this has to do with anything above,
other than the number 2 is involved.

> So I guessed at 256/255 (6.776 cents) because 256 is power of 2 and a few listening tests confirmed (to me at least) that it was an acceptable tempering value. Next I tried an 8 cent tempering and could just about detect a slight hint of dissonance (I'm pretty fussy) and so ruled it out. With dissonance I chose 16/15 (15 is one less than 16). Similarly I chose 256/255 because 255 is one less than 256. Think of 1/16 or 1/256.

OK. Well, that's how you came up with it. I'm not convinced that this
is how the ear works. If you'd like to keep exploring your system, go
right ahead. It would be nice if you tried to work your research in
with the rest of the research that has been done here.

> > Tempering 5/4 sharp by 14 cents sounds acceptable to most of the
> > western world, as does tempering 6/5 16 cents flat, so I'm not sure
> > your results apply.
>
> I worked out my formulas for *me* and not the western world. Sure you can "get away" with the mistunings you list above but I'm aiming for perfection or close enough to it.

Then your formulas apply to you and not the western world.

> > The only explanation that I have read is that this is the formula
> > you've picked because you like the results.
>
> This is partly true, the results do look "cute", magic numbers if you will. I guessed at hundreds of possible combinations of formulas and slowly eliminated them one by one (over a 14 year period making lots of mistakes) until I arrived at what I considered to be the most likely formula. It's not so much that "it must be formula 'a'" but rather "it can't be b or c or d or e or f or g or h etc".

More power to you, and you should always feel encouraged to do
research. The emerging theory that has come out of this list, which I
am very much a proponent of, is ongoing and has also spanned a period
of a decade and a half. It has so far included a world class
mathematician, several psychoacoustics experts, ethnomusicologists,
several outstanding composers and musicians, medieval music
historians, and so on. (I believe Carl claimed at one point a
neuroscientist was on here...?) I have been playing music since I was
two years old, and after having had teachers all my life and then
paying $50,000 to go to music school, I have never received a better
education in music theory at any point in my life than I have on this
list.

In light of this, I am very respectful of the work that has been done
so far and aim to try to contribute to it as best I can. I feel it is
of a high quality and actually makes sense in a way that many other
theories don't. Again, you should always feel encouraged to do your
own research, but there is no need to get upset if I also try to
inform newcomers of the drawbacks and advantages between different
models. And you should definitely not feel upset if I'm trying to help
you relate your model to existing models that have a bit more
explanatory power behind them.

> > There's also the fact that you haven't explained why a 12-tet major
> > chord sounds great to almost every single person living on the planet,
> > but contains no intervals that you'd mark "good."
>
> I never said that. In the 12TET open E Major chord on a guitar (E,B,E,G#,B,E) all the intervals that occur are within 6.776 cents of any of my own "good" intervals except the intervals that contain the G#. And why should it be my responsibility to explain this anyway?

It should be your responsibility to explain this because I thought the
point of your theory is that it predicts what will sound "good." And
my point is that 12-tet sounds so good that people have been driven
towards moments of enlightenment while listening to, playing, or
composing in it. Can't get much better than that.

-Mike

MikeB>"I don't care. 16/15 is used in chords all the time. OK, 12/11 is less harsh for critical bandwidth reasons. So what? To say that 16/15 is unusable as a harmonic interval is ridiculous."

   Of course 16/15 is usable, I never said it wasn't.  What I am saying, as far as semitone-like intervals go, there doesn't seem to be much evidence of 16/15 being "superior".
  I think there's a good deal of "zombie-ism" going on with people and 16/15 just because ratios near 16/15 happen to work well in 12TET-like chords that, unfortunately, makes such people happily skip right over the idea of using 12/11 as a semitone.
-------------------------------------
  Compare, for example 12:15:16:20  (Just C E F A) vs. 9:11:12:15.  Now, they are both good chords to my ears.  But , to me, the 9:11:12:15 has a fair advantage, though I'd find it interesting if someone thought the 12:15:16:20 had any significant advantage.

   Meanwhile if you go with clusters...10:11:12 or even 11:12:13 seems to have an uncanny advantage over 14:15:16 or 15:16:17.  There..as you mentioned, "critical band reasons" have a lot to do with it.

--------------------------
  If any of you all disagree with this...I would like to see some chords you believe proves 16/15 is "the (or perhaps even 'a') superior semitone".  I don't think it deserves its "monopoly" and that there are other, equally strong options.

On Sun, Mar 20, 2011 at 7:36 PM, Michael <djtrancendance@...> wrote:
>
>   Compare, for example 12:15:16:20  (Just C E F A) vs. 9:11:12:15.  Now, they are both good chords to my ears.  But , to me, the 9:11:12:15 has a fair advantage, though I'd find it interesting if someone thought the 12:15:16:20 had any significant advantage.

9:11:12:15 is also a nice chord, but it's just different. It's
12:15:18:22 in inversion. 12:15:16:20 is much brighter and happier.

>    Meanwhile if you go with clusters...10:11:12 or even 11:12:13 seems to have an uncanny advantage over 14:15:16 or 15:16:17.  There..as you mentioned, "critical band reasons" have a lot to do with it.

Sure.

>   If any of you all disagree with this...I would like to see some chords you believe proves 16/15 is "the (or perhaps even 'a') superior semitone".  I don't think it deserves its "monopoly" and that there are other, equally strong options.

I don't feel like 12/11 is a semitone.

-Mike

>
> The key advantage of Tenney is that it is diagonizable, by which I mean
> that 3/2 has the same height as 6, etc. It is, in essence, the L1 norm with
> weighted primes as the basis, and if you are willing to use the L2 norm
> instead, you obtain further mathematical advantages.
>

Right, I see what you mean. That was intuitively obvious from looking at the
lattices on the Tonalsoft pages, and also apparent from reading your
log-Tenney definition on the xenharmonic page, but every time you describe
it another way, I feel like it clicks a little better.

OK. TH rates 7/6 as better than 9/5, 9/7 as better than 11/6, 11/8 as better
> than 13/7, 11/10 as better than 15/8, 13/11 as better than 16/9, 15/13 as
> better than 18/11 and 16/13 as better than 20/11. From the point of view of
> Tenney height, O'Sullivan's measure is giving a small bonus to having
> smaller denominators and larger numerators, and so tends to give the nod to
> wider intervals.
>
> That reinforces my intuition, since the couple of intervals I had in mind
appear in that list. Thank you, that is just what I was looking for (not
reinforcement, per se, but helpful information along those lines).

Talking out loud here:

-lb( 2 + 1/x + 1/y - y/x )

= lb x + lb y - lb( (1-y) y + (1+2y) x )

This expression makes it more apparent to me just what is going on. For
"majorness", we want it ("surprisal", if you don't mind my calling it that)
to fall to 0 or less, so that

lb x + lb y <= lb( (1-y) y + (1+2y) x )

The left-hand side is log-Tenney. For large x,y, the right-hand side is
dominated by (2x-y) y, and there is a sort of volatile competition as the
function leaves the region where x=y/2. In this case, if x<y, then the
right-hand side is near to 0, meaning not "major". Otherwise, it is more
"major". This helps me understand a bit better what we are talking about.

MikeB>"I don't feel like 12/11 is a semitone."
   Fair enough.  In that case I must be a fan of the use of 7-tone scales without semitones. :-D

>"9:11:12:15 is also a nice chord, but it's just different. It's 12:15:18:22 in inversion. 12:15:16:20 is much brighter and happier."

  True...9:11:12:15 is undoubtedly darker.  Then again, I like smooth but minor-esque feeling chords.  Here use between the two comes down largely to subjectivity...

Me>    Meanwhile if you go with clusters...10:11:12 or even 11:12:13
seems to have an uncanny advantage over 14:15:16 or 15:16:17.  There..as
you mentioned, "critical band reasons" have a lot to do with it.

Mikeb>"Sure."

    This seems to beg the question do you or others think there would be good use for an Adaptive JI program that would push 15:16's in 12:11's when they occur?

   That way someone could, say, compose in 1/4 comma mean-tone...but have the program switch to a scale like the "Arab" mode of my infinity scale which replaces the 16/15 semitones for 12/11 when two notes a semi-tone apart are played at the same time.  This would hopefully make cluster chords a fair deal more reasonable to use...and hopefully the neutral intervals generated as a side effect would still serve well as the major/minor intervals played before the "switch".

On Sun, Mar 20, 2011 at 7:57 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"I don't feel like 12/11 is a semitone."
>    Fair enough.  In that case I must be a fan of the use of 7-tone scales without semitones. :-D
>
> >"9:11:12:15 is also a nice chord, but it's just different. It's 12:15:18:22 in inversion. 12:15:16:20 is much brighter and happier."
>
>   True...9:11:12:15 is undoubtedly darker.  Then again, I like smooth but minor-esque feeling chords.  Here use between the two comes down largely to subjectivity...

I don't think they're two intonations of the same chord.

> Me>    Meanwhile if you go with clusters...10:11:12 or even 11:12:13
> seems to have an uncanny advantage over 14:15:16 or 15:16:17.  There..as
> you mentioned, "critical band reasons" have a lot to do with it.
>
> Mikeb>"Sure."
>
>     This seems to beg the question do you or others think there would be good use for an Adaptive JI program that would push 15:16's in 12:11's when they occur?
>
>    That way someone could, say, compose in 1/4 comma mean-tone...but have the program switch to a scale like the "Arab" mode of my infinity scale which replaces the 16/15 semitones for 12/11 when two notes a semi-tone apart are played at the same time.  This would hopefully make cluster chords a fair deal more reasonable to use...and hopefully the neutral intervals generated as a side effect would still serve well as the major/minor intervals played before the "switch".

I guess you could use something like 33 or 40-equal, which is a
meantone with the fifths flat enough to generate a semitone of that
size. 26-equal has slightly better fifths and thirds, but the semitone
isn't in the range you want.

You will note that 26, 33, and 40 all differ by the number 7, and that
as you keep adding 7 you continually move the fifth asymptotically
closer to that of 7-equal, meaning you make it flatter. This is
because of magic. Similar magic will also make it sharper if you add
5.

-Mike

MikeB (concerning 9:11:12:15 vs. 12:15:16:20)>"I don't think they're two intonations of the same chord."

   Agreed...they are both different chords not different intonations...
  But I couldn't think of two closer chords (other than clusters) one of which uses 15/16 and the other of which uses 12/11....can you think of a more similar comparison where the odd-limit of the two chords is very similar?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> It's not a coincidence at all. 1/x + 1/y = (x+y)/(x*y). For large x
> and y, this will asymptotically approach Tenney Height.

Let's take the reciprocal of the O'Sullivan measure, so that they are both badness measures. If we set y=1, then Tenney height is x, which goes to infinity with increasing x; x/(x+1) goes to 1. If we set x=t+a, y=t-a then Tenney is t^2-a^2, which for fixed a goes to infinity quadratically with increasing x and y. O'Sullivan is (t^2-a^2)/(2t) = t/2 - a^2/(2t), both go to infinity, but not at the same rate.

Umm, ignore my last 3 sentences, please. (I blame distractions...)

On Sun, Mar 20, 2011 at 8:17 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > It's not a coincidence at all. 1/x + 1/y = (x+y)/(x*y). For large x
> > and y, this will asymptotically approach Tenney Height.
>
> Let's take the reciprocal of the O'Sullivan measure, so that they are both badness measures. If we set y=1, then Tenney height is x, which goes to infinity with increasing x; x/(x+1) goes to 1. If we set x=t+a, y=t-a then Tenney is t^2-a^2, which for fixed a goes to infinity quadratically with increasing x and y. O'Sullivan is (t^2-a^2)/(2t) = t/2 - a^2/(2t), both go to infinity, but not at the same rate.

Whoops, you're right. If x = y, then Tenney height is x^2, but
O'Sullivan height is 2x. My fault.

-Mike

What I meant to get at in the coda of my other post was..

Where x,y large and x = a y

the majorness condition is approx.

lb-Tenney <= lb( (2 a - 1) ) + 2 lb y

This is another illustration of what you mentioned, Gene, about favoring
larger numerators.

I made a mistake in the initial formula (forgot to divide by 2 inside the
log, or equivalently to subtract 1 outside the log)

I'll come back to this when I know what it is I'm trying to say.