Model for Interactions Between Harmonic Partials With Tempered Intervals?

Okay, Gene and Michael's objections to Harmonic Entropy as a model of concordance got me thinking. H.E. models interactions between fundamentals, but not harmonic partials. Is there any theory out there which can score how strongly partials will clash or not in a tempered interval, assuming an ideally-harmonic timbre? Did Sethares work this out? I'm starting to think that this would be more useful than Harmonic Entropy for evaluating the concordance of various EDOs. Any links or advice would be appreciated.

-Igs

Igs wrote:

> Okay, Gene and Michael's objections to Harmonic Entropy as a
> model of concordance got me thinking. H.E. models interactions
> between fundamentals, but not harmonic partials. Is there any
> theory out there which can score how strongly partials will
> clash or not in a tempered interval, assuming an ideally-harmonic
> timbre?

One can imagine higher-adic harmonic entropy where all the
partials are input. It would involve barycentric coordinates
in very high-dimensional spaces, and other such stuff I
don't understand. However Mike B. and I are working on what
may be kind of a shortcut. Stay tuned.

> Did Sethares work this out? I'm starting to think that
> this would be more useful than Harmonic Entropy for evaluating
> the concordance of various EDOs. Any links or advice would be
> appreciated.

Sethares has made available source code that computes the
roughness of an arbitrary complex of partials -- you can get
it from his website. However, the last time Paul tried it,
he got weird behavior when he did simple things like change
the amplitudes of all the partials. It is safe to say that
the field is ripe for the motivated individual to make a
contribution.

-Carl

Carl>"Sethares has made available source code that computes the roughness of an
arbitrary complex of partials -- you can get it from his website. However, the
last time Paul tried it, he got weird behavior when he did simple things like
change the amplitudes of all the partials."

Of course, because the way the entire program works is based on amplitudes.
I don't consider it weird at all. The program tries to align more influential
IE louder partials foremost. Thus when you start making partials far from the
root louder, you get something that looks less and less like the HE curve.
When Igs said "H.E. models interactions between fundamentals, but not
harmonic partials", this brings into perspective the idea that HE focuses more
on root-tone periodicity and alignment while Sethares theory can also lean more
toward non-root-tone alignment. The catch is...Sethares' "proof of JI"
examples assume a timbre that slopes down heavily and focuses on the root tone
(much like that of many acoustic instruments!)...and the side effect is that
such examples form a curve which looks much like the HE curve.

Now the real question, IMVHO, is where is the fine balance between overtone
alignment and root tone alignment AKA periodicity? Sure...if you have to only
use one theory I think we can almost all agree periodicity is better...but the
obvious observation seems to be they both matter and it's never a case of
completely one and not the other.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
>
> Of course, because the way the entire program works is based on amplitudes.
> I don't consider it weird at all. The program tries to align more influential
> IE louder partials foremost. Thus when you start making partials far from the
> root louder, you get something that looks less and less like the HE curve.
> When Igs said "H.E. models interactions between fundamentals, but not
> harmonic partials", this brings into perspective the idea that HE focuses more
> on root-tone periodicity and alignment while Sethares theory can also lean more
> toward non-root-tone alignment. The catch is...Sethares' "proof of JI"
> examples assume a timbre that slopes down heavily and focuses on the root tone
> (much like that of many acoustic instruments!)...and the side effect is that
> such examples form a curve which looks much like the HE curve.

I think the biggest discrepancy seems to happen around the fifth, since on many acoustic instruments, the 3rd partial is pretty loud. Come to think of it, 12-tET makes a weird sort of sense when you look at the amplitude of partials on acoustic instruments, in that the "in-tune-ness" of various notes practically seems correlated to volume of partials! I mean, the octave is pure, and the 2nd partial is always very strong, the fifth is very close to pure, and the 3rd partial is quite strong, then the major third is not very pure, and the 5th partial is not super-strong.

So one of the paradoxes is that 3/2 is very periodic and has low Tenney height, so being near it will (in the sense of fundamentals) theoretically sound concordant, but when you look at the 3rd partial of the fundamental vs. the 2nd partial of the fifth, they'll start to clash pretty heavily as you mistune the fifth, because the interval between THEM will quickly rise in harmonic entropy and/or critical band roughness, and/or decrease in periodicity. Hence a mistuned fifth smuggles in really HIGH H.E. intervals between partials.

Interesting.

-Igs

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

> One can imagine higher-adic harmonic entropy where all the
> partials are input. It would involve barycentric coordinates
> in very high-dimensional spaces, and other such stuff I
> don't understand. However Mike B. and I are working on what
> may be kind of a shortcut. Stay tuned.

Couldn't you just use a formulation that looks at the H.E. of the intervals between each pair of partials (using up to maybe the first 8 partials) between the two notes, weights them according to the amplitude of the partials, and then creates an appropriate average? It wouldn't be perfect, but might not it be good enough?

> Sethares has made available source code that computes the
> roughness of an arbitrary complex of partials -- you can get
> it from his website. However, the last time Paul tried it,
> he got weird behavior when he did simple things like change
> the amplitudes of all the partials. It is safe to say that
> the field is ripe for the motivated individual to make a
> contribution.

I'll have to check it out. Though I don't really know what to do with source code.

-Igs

Michael wrote:

> Of course, because the way the entire program works is based
> on amplitudes. I don't consider it weird at all.

What don't you consider weird? Because I didn't say which
behaviors he noticed. And it looks like you're just writing
nonsense again.

-Carl

Igs wrote:

> > One can imagine higher-adic harmonic entropy where all the
> > partials are input. It would involve barycentric coordinates
> > in very high-dimensional spaces, and other such stuff I
> > don't understand. However Mike B. and I are working on what
> > may be kind of a shortcut. Stay tuned.
>
> Couldn't you just use a formulation that looks at the H.E.
> of the intervals between each pair of partials

Sure, and Paul did, and we recently discussed the pros and
cons of this pairwise formulation, did we not?

> (using up to maybe the first 8 partials) between the two notes,
> weights them according to the amplitude of the partials, and
> then creates an appropriate average?

Paul didn't weight by amplitude, so that would be something
interesting to try.

-Carl

Me> Of course, because the way the entire program works is based
> on amplitudes. I don't consider it weird at all.

Carl>What don't you consider weird? Because I didn't say which
>behaviors he noticed. And it looks like you're just writing
>nonsense again.

Before you just said that Paul said the results seemed "weird". Vague
statement. And I replied (based on what admittedly little you had given as a
topic guideline) why I thought they were not weird. And, yes, I gave a
guesstimate on how I took it that the results could be interpreted as weird
based on both Mike B's statements about what HE does (vs. what people think it
does) and does not summarize vs. critical band dissonance and my own experience
with the two theories. Not once did I say "what Paul said is wrong" I simply
said "I don't see a reason why it would be weird if you understand Critical Band
Dissonance focuses more on "on the average" overtone alignment and HE generally
more on root-tone periodicity.

>"And it looks like you're just writing nonsense again."
if you make a vague statement about what Paul said and then get frustrated
with my lack of ability to guess what the details were behind your vague
statement....well (duh?) maybe you should have gave more details on exactly
what Paul said and thus avoid making such a vague statement! :-S
So if it's really so important to you (and you're able to actually back your
vague claim up)...what SPECIFICALLY did Paul say about the results from
Sethares' program being weird?

Michael wrote:

> And, yes, I gave a guesstimate on how I took it that the
> results could be interpreted as weird

Well, thanks for joining in the discussion to pontificate
on reasons why it might be that results you know nothing
about might be considered weird. Very helpful!

Paul found that Sethares' algorithm is majorly broken, or
at least, he found major evidence of majorly pathological
behavior that Bill declined to address when asked.
I discussed the issue in some detail on the list this
year already.

> if you understand Critical Band Dissonance focuses more on
> "on the average" overtone alignment and HE generally
> more on root-tone periodicity.

I'm not the one suffering from a lack of understanding at
this moment.

> if you make a vague statement about what Paul said and
> then get frustrated with my lack of ability to guess what
> the details were behind your vague statement....

You may have found my statement vague, but then again, I
was replying to someone else. If you wanted clarification,
why not ask?

> what SPECIFICALLY did Paul say about the results from
> Sethares' program being weird?

There, that wasn't so hard, was it?

/tuning/topicId_85909.html#86434
/tuning/topicId_85909.html#86441
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-Carl

On Mon, Sep 6, 2010 at 10:18 PM, Michael <djtrancendance@...> wrote:
>
>    Along these lines (let me get this right): Igs, Mike B, and my own doubts are all completely wrong and your and Paul's HE Pride (sounds kind of like a high school cheer squad) are all right?   Sounds like you're not even up for debating and analyzing...and that you already chose your/"the" right answer years ago.  Boo.

I think HE is great for what it's great for. When I started looking
into alternative ways to formulate it, I started to see some of the
ways it could be improved upon. I would never take any model of any
psychoacoustic property and let it override my own judgment on what's
consonant or not. That being said, HE has a purpose in mind, and it's
a monumental step in the right direction. Carl and I are working on a
way to extend and reformulate it, which will hopefully prove to be a
further step in the right direction. I have proposed another way to
extend (and reformulate it) on tuning-math, which will also hopefully
prove to be a further further step in the right direction.

That is, I don't really have conceptual doubts about HE, but I do have
doubts that it takes precedence over my own judgement. It's not
perfect, but it's the best thing we have, and for what it's worth it
can be immensely useful for something like what Igs wants to do. I
would encourage you to come up with your own "multi-consonance"
formula, however, if you want to come up with your own consonance
measure for complex timbres. So you think 60% is HE and 40% is beating
partials? Work the math out and do it! We can use all of the research
that we can get.

That is to say, you sometimes get attacked on this list. I think that
it isn't so much because your ideas diverge from "the norm," but
rather that you tend to debate the norm rather than coming up with
your own models. I think you have come up with some novel and creative
ideas since you joined this list. Put some original research out and I
would imagine that you'd find some receptive ears on here.

-Mike

>"I think that it isn't so much because your ideas diverge from "the norm," but
>rather that you tend to debate the norm rather than coming up with your own
>models."
Perhaps I haven't been clear enough. I, for example, tried my best to say
that I think critical band should be used more often with dyadic ratios under
around 7/6....implying that I think it should almost always be used in such
cases. Then I tried giving examples of the (to my ear) flat range (so far as
perceived dissonance) between about 12/11 and 9/8...and Igs said that 9/8 was
significantly better IE no dice.

Maybe I should be asking...what degree of examples do I need to provide to
get something around here taken as a theory? I feel I keep running into a
brick wall when someone asks me "ok, what existing theory are you following".
In such cases I can't find "PHD research" to back up things. Like my ideas of
using clustered triads with consecutive slightly tempered neutral seconds as
chords or using 22/15 as a 5th occasionally and trying to balance/"re-align" the
impure 5th with generally strong dyads. I've given lots of sound examples (both
plans for sound experiments/reproduction and actual sound samples)...but those
don't seem to be counting much either.

>"So you think 60% is HE and 40% is beating partials? Work the math out and do
>it! We can use all of the research
that we can get."

I feel in the same boat as you in that, in some cases, my ear will tell me
something that seems to contradict many existing theories and I'll trust my
ear. While I can't give a series that explains the 60/40....looking at the two
curves seems to say the following. At least to my ear...anything under about
7/6 follows Sethares' curve more closely (+1 point). Sethares' curve also puts
6/5,5/4, 5/3, and 4/3 (+4 points) at much more similar dissonance levels than
HE...just like my ears do. So make that 5 points for critical band dissonance.

The HE curve shows a high discordance point at around 22/15 and Sethares'
curve shows the area as fairly level....whereas to my ears the point at 22/15 is
much more settled than anything around it....and specifically more settled than
16/11, which looks like a relative low-point on the HE curve. Plus they both
miss the 11/9 neutral second, the 18/11's sounding much more consonant to me
than 13/8, the 11/6's sounding much more consonant than the nearby 20/11...and a
handful of other "neutral interval relative discordance low points".

IMVHO, both curves fail there, but HE fails a good bit worse. Same goes with
the way both systems seem to over-credit 3/2 and give, say, 40/27 a fairly good
rating just because it nears 3/2. 0 points for both theories.

Pros for HE: it catches the low point around 436 cents (about 9/7), one at
5/4 (giving a wider low "gravity" range than Sethares' curve), another definite
low around 7/5, another at 8/5, another at 7/4...significantly better than
Sethares. A point that applies to all 5 of these, I can really sense the
"gravity" toward those points....and something near 5/4, for example, often just
sounds like a bad 5/4 and not a good something else. For example 24/19 between
5/4 and 9/7...it sounds more like 5/4 than 9/7 because of such gravity. So make
that 5*1.5 (for the overlying gravity concept and said above cases where I find
it works well) = +7.5 for HE.

And since 7.5 + 5 = 12.5 and 7.5 / 12 = 60%...there you get your 60% that HE is
better.
Now so far as what to do productively with the 22/15 and 9/8 12/11 range
that I say both theories miss the mark on...oh man. My composition for Sevish's
project uses 18/11, 11/6, 22/15...all over the place and I've posted tons of
suggested scales using those intervals. I've also given several examples of
"super diminished" chords I think are passable discordance-wise like stacking a
12/11 * 11/10 into a 6/5. The usual response is "no it doesn't...because said
established theory says it doesn't"...but hey, at least I try.

>"Put some original research out and I would imagine that you'd find some
>receptive ears on here."
Again...any suggestions of what I could do to better qualify my efforts as
"original research"? I'm all ears... :-)

On Tue, Sep 7, 2010 at 12:00 AM, Michael <djtrancendance@...> wrote:
>
> >"I think that it isn't so much because your ideas diverge from "the norm," but rather that you tend to debate the norm rather than coming up with your own models."
>     Perhaps I haven't been clear enough.  I, for example, tried my best to say that I think critical band should be used more often with dyadic ratios under around 7/6....implying that I think it should almost always be used in such cases.  Then I tried giving examples of the (to my ear) flat range (so far as perceived dissonance) between about 12/11 and 9/8...and Igs said that 9/8 was significantly better IE no dice.

Right, you have said that. And I have repeatedly thrown in counter
examples in which chords are used that have intervals smaller than
7/6, and they sound great. You then insist that they don't sound
great. For whatever reason, rather than coming to the realization that
roughness is a phenomenon that clearly has some subjective tolerance
component to it - you insist that intervals less than that are
unusable. You will never manage to prove that roughness "is unpleasant
to listen to," because lots of people seem to like it.

>    Maybe I should be asking...what degree of examples do I need to provide to get something around here taken as a theory?   I feel I keep running into a brick wall when someone asks me "ok, what existing theory are you following".  In such cases I can't find "PHD research" to back up things.   Like my ideas of using clustered triads with consecutive slightly tempered neutral seconds as chords or using 22/15 as a 5th occasionally and trying to balance/"re-align" the impure 5th with generally strong dyads.  I've given lots of sound examples (both plans for sound experiments/reproduction and actual sound samples)...but those don't seem to be counting much either.

There is no PhD research, because there are no PhD's here. If you have
a theory for how musical consonance works, it needs to hold in almost
all cases, and the exceptions need to be adequately addressed. If it
seems useful, people will use it. Your ideas about critical band
dissonance in chords seem to be you expressing articulately your own
personal tastes in the construction of harmonies, not a theory for the
foundations of music. But if lots of people are saying they prefer
roughness in chords, and you say that you don't - isn't it a bit
simplistic to continue to argue over whether or not roughness is
"bad?" Clearly some people prefer it and some don't.

>     I feel in the same boat as you in that, in some cases, my ear will tell me something that seems to contradict many existing theories and I'll trust my ear.   While I can't give a series that explains the 60/40....looking at the two curves seems to say the following.  At least to my ear...anything under about 7/6 follows Sethares' curve more closely (+1 point).  Sethares' curve also puts 6/5,5/4, 5/3, and 4/3 (+4 points) at much more similar dissonance levels than HE...just like my ears do.  So make that 5 points for critical band dissonance.
//
> IMVHO, both curves fail there, but HE fails a good bit worse. Same goes with the way both systems seem to over-credit 3/2 and give, say, 40/27 a fairly good rating just because it nears 3/2. 0 points for both theories.

Right, but the point that keeps being made here is that these two
dissonance measures aren't just two mysterious curves that either
match up to or differ with perception. It isn't like this is politics,
with there being the Erlich and the Sethares party and people figuring
out which cult figure they like more. The two curves have conceptual
(and mathematical) ideas behind them for one to think critically
about. The conceptual idea behind the critical band curve is that
roughness causes dissonance, and everything sort of follows from that.
So the question is - what about if we're dealing with sine waves? Is
everything equally consonant outside of the critical band range?

Harmonic entropy and Sethares dissonance simply deal with two
different psychoacoustic phenomena. To take either one as being
responsible for the entire concept of consonance is a bit simplistic -
so why do it? Keep again that a bare major 7th dyad sounds consonant
in a way that isn't described by either HE or critical band
dissonance.

>    Pros for HE: it catches the low point around 436 cents (about 9/7), one at 5/4 (giving a wider low "gravity" range than Sethares' curve), another definite low around 7/5, another at 8/5, another at 7/4...significantly better than Sethares.  A point that applies to all 5 of these, I can really sense the "gravity" toward those points....and something near 5/4, for example, often just sounds like a bad 5/4 and not a good something else.  For example 24/19 between 5/4 and 9/7...it sounds more like 5/4 than 9/7 because of such gravity.  So make that 5*1.5 (for the overlying gravity concept and said above cases where I find it works well) = +7.5 for HE.
>
> And since 7.5 + 5 = 12.5 and 7.5 / 12 = 60%...there you get your 60% that HE is better.

Right, so what you're doing is trying to come up with an overall
"pleasantness" concept that factors in how much the interval is
tempered and how much the partials beat, for harmonic timbres. There's
nothing wrong with that. I don't see the rigor in your approach by
assigning arbitrary numbers as "points" and then using those as the
basis for a mathematical calculation, but you are free to do whatever
you wish and if you find the end result useful, use it.

>     Now so far as what to do productively with the 22/15 and 9/8 12/11 range that I say both theories miss the mark on...oh man.  My composition for Sevish's project uses 18/11, 11/6, 22/15...all over the place and I've posted tons of suggested scales using those intervals.  I've also given several examples of "super diminished" chords I think are passable discordance-wise like stacking a 12/11 * 11/10 into a 6/5.  The usual response is "no it doesn't...because said established theory says it doesn't"...but hey, at least I try.

I have always thought your scales sound great. I don't think I
understand exactly why you think they sound great. I do agree that
there is some kind of "sweet spot" around the neutral second, and I
don't know why exactly that is. I suspect it has something to do with
roughness.

> >"Put some original research out and I would imagine that you'd find some receptive ears on here."
>  Again...any suggestions of what I could do to better qualify my efforts as "original research"?   I'm all ears... :-)

I responded to this offlist with a longwinded rant.

-Mike

MikeB>"Right, you have said that. And I have repeatedly thrown in counter
examples in which chords are used that have intervals smaller than
7/6, and they sound great."
Perhaps there's been some lack of clarity on my part. :-( I was saying that
the critical band dissonance curve of Sethares gives what I see as more accurate
estimates IE it seems to shoot up in dissonance much more starting at about
13/12 (and anything below that has a much lower incremental penalty for
increased closeness)...whereas HE seems to say there is more or less a linear
increase in dissonance starting at around 7/6 (boo...thanks to my ears...I don't
believe that for a second). :-D

If you really think I'm against intervals less than 7/6...look at my
composition for Sevish's album, look at almost any of my latest scales
(including the Infinity-series scales) which as chalk full of 12/11 intervals or
scales I favor ala Ptolemy's Homalon scales, chords I've recommended,
suggestions I gave to Jon (suggestion he allow 10/9 if not also 11/10 and 12/10
to be counted as chord-usable dyads in his consonance calculator...) And
check how many times I've used the keywords "neutral second" in my messages on
this list in a positive light....it could easily be hundreds of times!

I'm about the largest SUPPORTER of using consecutive small intervals as you
will find....minus the kind of people who stack consecutive near 16/15 (or
closer!) dyads and call the result a stable chord.

>"you insist that intervals less than that are unusable."
Erm...not at all. I'd say intervals less than 12/11 are fairly unusable in
chords. That is unless...they are separated by larger intervals to compensate
for their dissonance IE D E F in 12TET seems OK because the D is about 9/8 from
the E, which helps make up for the terrible critical band dissonance of the
E->F.

>"Your ideas about critical band dissonance in chords seem to be you expressing
>articulately your own
personal tastes in the construction of harmonies, not a theory for the
foundations of music."
And now that I've clarified where I think the limits of chord density are
(more like 12/11, certainly not at 7/6!)...would you still say that?

>"But if lots of people are saying they prefer roughness in chords, and you say
>that you don't"
Well, some people prefer Sethares "roughness based", some don't. So wouldn't
that idealogy you seem to be charging against my ideas with throw his ideas out
the window as well?

>"isn't it a bit simplistic to continue to argue over whether or not roughness is
>"bad?" Clearly some people prefer it and some don't."
Argh. I know a few professional musicians. I've played them micro-tonal
chords and asked them to pick out what sounds bad to them. Without fail, they
jumped at consecutive 12TET semi-tone-like intervals...a couple even jumped at
my precious consecutive minor second chords (though much less so). Some even
found, by ear, tail ends of melodies (not very sustained) that combined with
each other to form high critical band dissonance tones I didn't even know were
in my compositions. Listeners...same deal...only their ears didn't pick up
things like neighboring tone conflicts so accurately. Is it all-encompassing?
No. Does it appear to apply/work more than half the time far as people I've
tested it on? You bet!

>"Right, but the point that keeps being made here is that these two dissonance
>measures aren't just two mysterious curves that either match up to or differ
>with perception. "
Meaning...there's a mathematical principle behind each one? If so, yes, I get
that. But then again Sethares was based on Plomp and Llevelt...and their curve
was derived by a series of tests involving two sine waves over many people just
asking "how dissonant is this?"

>"Right, so what you're doing is trying to come up with an overall "pleasantness"
>concept that factors in how much the interval is tempered and how much the
>partials beat, for harmonic timbres."
I just don't see how that's any more "random" than what Plomp and Levelt
did...specifically if I can find enough listeners to give evidence to back it up
(as they did). Otherwise...it seems the only path to a "justified theory" is
to make every single little thing fit into one hulking formula that assigns
exact values to everything...is that really what's required?

>"I have always thought your scales sound great. I don't think I understand
>exactly why you think they sound great."

>"I do agree that there is some kind of "sweet spot" around the neutral second"
Mysteriously...yes...although perhaps more of a sweet range than a sweet
spot. :-)
It seems the feel and "usability" between 12/11 and 9/8 is indeed locked...to
the point the root tone beating becomes significant enough that periodicity
doesn't matter as much...this adds to the fact most everything is within 7 cents
of everything else in that range anyhow. Try 23/21 vs. 11/10...hard to tell a
difference despite the 23/21 being higher limit. My point...(consistently)
under about 9/8 periodicity becomes "blurred"...which means the distinguishing
factor left is critical band. And, even with pure sine waves, the critical band
under 12/11 seems to be more than the ear can handle.

>"and I don't know why exactly that is. I suspect it has something to do with
>roughness."
Right...or a least I'm trying to "play a game" of getting as many neutral
seconds as possible to create new kinds of clustered chords without destroying
the accuracy of more common ratios (IE the low-limit ones HE points out). I am
using the apparent fact that range sounds "blurred" to round with arbitrary
precision between 12/11 and 9/8 to help preserve those larger ratios (IE the
ratios where periodicity quite often takes precedence). Also the latest scales
I've made capture all possible dyads within a 2 octave range to a list of
desirable dyads (mostly trying to form 7-limit or less allowing a few of the
more relaxed sounding 11-limits as needed) within 8 cents...IE the whole
"maxi-min" idealism from 1/4 comma mean-tone definitely seems to help.

Some things I can't quite explain yet...like how I've found sour areas near
16/13 and 20/11 and 16/11 and 13/8....and yet sweet areas around 22/15 and 13/9
and 18/11 (which look on the surface to be just as bad as fractions). Finding
a unified theory to explain those will take a while...all I can say for now is
those appear to be some very nearby bad-next-to-good ranges and the musicians
I've blind tested those dyads on seem to agree a large majority of the time so
far.

I wrote:
> There is no PhD research, because there are no PhD's here.

Haha, I'm getting messaged offlist because of this line. I should
clarify: of course there are people here who possess PhD's in various
things. But a lot of the theory coming out of this list originated on
this list and away from academia, is more what I was getting at. In
the world of most PhD's, all of this doesn't exist.

-Mike

On Tue, Sep 7, 2010 at 2:31 AM, Michael <djtrancendance@...> wrote:
>
> MikeB>"Right, you have said that. And I have repeatedly thrown in counter
> examples in which chords are used that have intervals smaller than
> 7/6, and they sound great."
>   Perhaps there's been some lack of clarity on my part. :-(  I was saying that the critical band dissonance curve of Sethares gives what I see as more accurate estimates IE it seems to shoot up in dissonance much more starting at about 13/12 (and anything below that has a much lower incremental penalty for increased closeness)...whereas HE seems to say there is more or less a linear increase in dissonance starting at around 7/6 (boo...thanks to my ears...I don't believe that for a second). :-D

HE says that there is a linear increase in the number of perspectives
that your brain can assume for a certain dyad, if that dyad is played
by itself and with sine waves. Nothing more. You cannot generalize the
HE curve to hold for triads, tetrads, and so on. To give an obvious
example, 16/15 is considered dissonant on the curve, but it's clearly
consonant within the context of 1:2:3:4:5:6:7:8:...:14:15:16, where it
forms the most consonant chord there is.

>     I'm about the largest SUPPORTER of using consecutive small intervals as you will find....minus the kind of people who stack consecutive near 16/15 (or closer!) dyads and call the result a stable chord.

It is difficult to stack two minor seconds on top of each other and
come up with a consonant chord. I don't think impossible though.

>    Erm...not at all.   I'd say intervals less than 12/11 are fairly unusable in chords.  That is unless...they are separated by larger intervals to compensate for their dissonance IE D E F in 12TET seems OK because the D is about 9/8 from the E, which helps make up for the terrible critical band dissonance of the E->F.

Fair enough, but I think it's possible to come up with a Godel
statement for that paradigm as well.

> >"Your ideas about critical band dissonance in chords seem to be you expressing articulately your own
> personal tastes in the construction of harmonies, not a theory for the foundations of music."
>     And now that I've clarified where I think the limits of chord density are (more like 12/11, certainly not at 7/6!)...would you still say that?

I think that I've gone through different phases myself where I try to
figure out "the rules," and then I inevitably start to feel restricted
and find a way to break the rules. That being said, placing two half
steps in a row is certainly a very dissonant sound, but I wouldn't
consider it unusable for functional harmony.

>    Well, some people prefer Sethares "roughness based", some don't.  So wouldn't that idealogy you seem to be charging against my ideas with throw his ideas out the window as well?

I'm not sure if Sethares uses his curve to postulate some kind of
"cutoff," above which the roughness would render a dyad or chord
completely unusable. If he does, I would disagree with that, although
his curve would certainly still have some use.

>    Argh.   I know a few professional musicians.  I've played them micro-tonal chords and asked them to pick out what sounds bad to them.  Without fail, they jumped at consecutive 12TET semi-tone-like intervals...a couple even jumped at my precious consecutive minor second chords (though much less so).  Some even found, by ear, tail ends of melodies (not very sustained) that combined with each other to form high critical band dissonance tones I didn't even know were in my compositions.  Listeners...same deal...only their ears didn't pick up things like neighboring tone conflicts so accurately.  Is it all-encompassing?  No.  Does it appear to apply/work more than half the time far as people I've tested it on?  You bet!

I'm a professional musician who's also into microtonal music, and I'd
like to hear some of these tail ends of melodies that cause so much
critical band dissonance :)

>   Meaning...there's a mathematical principle behind each one?  If so, yes, I get that.  But then again Sethares was based on Plomp and Llevelt...and their curve was derived by a series of tests involving two sine waves over many people just asking "how dissonant is this?"

There are serious conceptual differences between each one, which is
also the point. They aren't referring to the same psychoacoustic
thing.

>      I just don't see how that's any more "random" than what Plomp and Levelt did...specifically if I can find enough listeners to give evidence to back it up (as they did).   Otherwise...it seems the only path to a "justified theory" is to make every single little thing fit into one hulking formula that assigns exact values to everything...is that really what's required?

I haven't looked at Plomp and Levelt into too much detail, you'll have
to reference me some stuff before I can really comment.

> My point...(consistently) under about 9/8 periodicity becomes "blurred"...which means the distinguishing factor left is critical band.  And, even with pure sine waves, the critical band under 12/11 seems to be more than the ear can handle.

Didn't you just say that you were an advocate of minor seconds in chords??

-Mike

Me> My point...(consistently) under about 9/8 periodicity becomes
"blurred"...which means the distinguishing factor left is critical band. And,
even with pure sine waves, the critical band under 12/11 seems to be more than
the ear can handle.
mikeB>"Didn't you just say that you were an advocate of minor seconds in
chords?"
Nah, I said neutral seconds. :-D In the case of neutral seconds, btw, I
consider "blurred" to be a good thing...it means around that range I can temper
off an JI ratio at random and not be "penalized" for it. Hence in my new
scales, I keep a very close eye on the periodicity of anything of about 6/5 and
up....but happily let anything between 9/8 and 12/11 vary off periodic values
because, again, I strongly suspect that within that range sense of periodicity
becomes blurred and critical band dissonance begins to take over more as
intervals get smaller beyond that point.

Of course, that part of my theory would have to hold for minor seconds (IE
around 16/15) too...even if I don't like them so much...since they are also part
of the "blurred" intervals of about 9/8 and under. So, to me, deciding if an
16/15 ruins a would be 10:14:15 chord becomes irrelevant as a "blurred" 16/15
would likely sound almost indistinguishable from the 15/14.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > Couldn't you just use a formulation that looks at the H.E.
> > of the intervals between each pair of partials
>
> Sure, and Paul did, and we recently discussed the pros and
> cons of this pairwise formulation, did we not?

Well, we discussed using it as a model for higher-adic harmony, and it was clear that it didn't work for that because it fails to take note order into account. What I'm suggesting is not really "higher-adic" since we're still dealing with dyads, though it looks higher-adic because we're making calculations based on multiple partials. But in this case "ordering" is not an issue because a harmonic series ordering of partials is assumed from the get-go. And at any rate, ordering isn't really important, what is important is the amplitude of any high-entropy intervals between partials. It only matters which two partials insofar as lower partials are louder partials, and loudness is taken into account by weighting-by-amplitude.

Now, I wish I had any idea how to actually do these calculations. Let's see...to generate the list of intervals formed between pairs of the first 8 partials of both notes in a dyad, you'd start with the lower note (i.e. 0 cents) and list the cents-values of its partials. This gives you List A, A1-A8. Then, whatever the dyad is that you're evaluating--say 675 cents--you add that to each member of List A (A1-A8) to give your set of partials for the higher note. This gives you List B (B1-B8). Then to find all the intervals between them, you subtract whichever number is lower in each list from whichever number is higher. This should give you, what, 64 intervals? That's as far as I can get. I have no idea how to apply weighting of any sort. Carl? Gene? Graham? Anyone? Or is this idea not accurate enough to be worth pursuing?

-Igs

I would suggest doing it without (what I'm guessing) you mean by weighting.
What I think you mean by weighting would be how to account for the harmonic
spectrum of each fundamental note.

Chris

On Tue, Sep 7, 2010 at 10:37 AM, cityoftheasleep <igliashon@...>wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Carl Lumma"
> <carl@...> wrote:
> > > Couldn't you just use a formulation that looks at the H.E.
> > > of the intervals between each pair of partials
> >
> > Sure, and Paul did, and we recently discussed the pros and
> > cons of this pairwise formulation, did we not?
>
> Well, we discussed using it as a model for higher-adic harmony, and it was
> clear that it didn't work for that because it fails to take note order into
> account. What I'm suggesting is not really "higher-adic" since we're still
> dealing with dyads, though it looks higher-adic because we're making
> calculations based on multiple partials. But in this case "ordering" is not
> an issue because a harmonic series ordering of partials is assumed from the
> get-go. And at any rate, ordering isn't really important, what is important
> is the amplitude of any high-entropy intervals between partials. It only
> matters which two partials insofar as lower partials are louder partials,
> and loudness is taken into account by weighting-by-amplitude.
>
> Now, I wish I had any idea how to actually do these calculations. Let's
> see...to generate the list of intervals formed between pairs of the first 8
> partials of both notes in a dyad, you'd start with the lower note (i.e. 0
> cents) and list the cents-values of its partials. This gives you List A,
> A1-A8. Then, whatever the dyad is that you're evaluating--say 675 cents--you
> add that to each member of List A (A1-A8) to give your set of partials for
> the higher note. This gives you List B (B1-B8). Then to find all the
> intervals between them, you subtract whichever number is lower in each list
> from whichever number is higher. This should give you, what, 64 intervals?
> That's as far as I can get. I have no idea how to apply weighting of any
> sort. Carl? Gene? Graham? Anyone? Or is this idea not accurate enough to be
> worth pursuing?
>
> -Igs
>
>
>

Igs wrote:

> > > Couldn't you just use a formulation that looks at the H.E.
> > > of the intervals between each pair of partials
> >
> > Sure, and Paul did, and we recently discussed the pros and
> > cons of this pairwise formulation, did we not?
>
> Well, we discussed using it as a model for higher-adic harmony,
> and it was clear that it didn't work for that because it fails
> to take note order into account. What I'm suggesting is not
> really "higher-adic" since we're still dealing with dyads,
> though it looks higher-adic because we're making calculations
> based on multiple partials. But in this case "ordering" is not
> an issue because a harmonic series ordering of partials is
> assumed from the get-go.

If you're willing to assume dyads of harmonic timbres it might
work, though it is still possible to construct things like
subharmonic chords. For instance, we can make 1/1 5/4 10/7 5/3
with the dyad 21/20 (use fixed-width font):

1 2 3 4 5 6 7 8
1/1 5/4
10/7 5/3
1 2 3 4 5 6 7 8

The upper tone provides 1/1 & 5/4 with its 4th & 5th partials
while the lower tone provides 10/7 and 5/3 with its 6th & 7th
partials. If you let me use two different timbres I can make
this perfect by boosting the appropriate partials. Having to
use a single timbre might still produce a weird result if the
4th-7th partials were boosted.

> Or is this idea not accurate enough to be worth pursuing?

It's a fine idea though I'm going to pursue something else.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > Well, we discussed using it as a model for higher-adic harmony,
> > and it was clear that it didn't work for that because it fails
> > to take note order into account. What I'm suggesting is not
> > really "higher-adic" since we're still dealing with dyads,
> > though it looks higher-adic because we're making calculations
> > based on multiple partials. But in this case "ordering" is not
> > an issue because a harmonic series ordering of partials is
> > assumed from the get-go.
>
> If you're willing to assume dyads of harmonic timbres it might
> work,

Yes, the *point* is that we are assuming dyads of harmonic timbres. It's not a universal or a general formulation, just something to give a rough guide.

> though it is still possible to construct things like
> subharmonic chords. For instance, we can make 1/1 5/4 10/7 5/3
> with the dyad 21/20 (use fixed-width font):
>
> 1 2 3 4 5 6 7 8
> 1/1 5/4
> 10/7 5/3
> 1 2 3 4 5 6 7 8
>
> The upper tone provides 1/1 & 5/4 with its 4th & 5th partials
> while the lower tone provides 10/7 and 5/3 with its 6th & 7th
> partials. If you let me use two different timbres I can make
> this perfect by boosting the appropriate partials. Having to
> use a single timbre might still produce a weird result if the
> 4th-7th partials were boosted.

Sorry, I don't really get what you're doing here. But that's okay, because the more I think about it, the more I think this idea isn't really worth pursuing. I mean, I'm looking for some quantifiable way to easily compare the concordance of EDOs along a single dimension. I thought H.E. was the way to go for a while, but now I'm not entirely sure. I mean, it works great for sine waves, but for harmonic timbres the interactions between partials will play a stronger role in determining concordance (or "sensory consonance" if that's a better term). I mean, with a sawtooth wave or a distorted guitar, there's no way 400 cents is going to sound smoother/more relaxed than 320 cents, but 400 cents has lower H.E. than 320 cents. 720 cents has lower H.E. than both, so I just don't think H.E. is going to work out as a reliable guide to concordance, even though I would think it should.

Perhaps I should give up on the idea that different EDOs can be quantitatively compared along a single dimension, or that it's even worthwhile to quantify qualitative properties of sound? Maybe I can get the point across about what each EDO is and isn't good for just by words.

-Igs

On Tue, Sep 7, 2010 at 4:12 PM, cityoftheasleep <igliashon@...> wrote:
> Sorry, I don't really get what you're doing here. But that's okay, because the more I think about it, the more I think this idea isn't really worth pursuing. I mean, I'm looking for some quantifiable way to easily compare the concordance of EDOs along a single dimension. I thought H.E. was the way to go for a while, but now I'm not entirely sure. I mean, it works great for sine waves, but for harmonic timbres the interactions between partials will play a stronger role in determining concordance (or "sensory consonance" if that's a better term). I mean, with a sawtooth wave or a distorted guitar, there's no way 400 cents is going to sound smoother/more relaxed than 320 cents, but 400 cents has lower H.E. than 320 cents. 720 cents has lower H.E. than both, so I just don't think H.E. is going to work out as a reliable guide to concordance, even though I would think it should.

It works out as a reliable guide to a certain type of concordance. I
have this feeling the curve would look very, very different if it were
calculated with harmonic timbres instead of sine waves. Small
mistunings would produce huge increases in entropy, I think, since
you'd start seeing all kinds of 30-50 cent dyads popping up in the
upper partials.

> Perhaps I should give up on the idea that different EDOs can be quantitatively compared along a single dimension, or that it's even worthwhile to quantify qualitative properties of sound? Maybe I can get the point across about what each EDO is and isn't good for just by words.

Nonsense! All you need is some kind of "badness" measure. Something
that takes into account the following factors:
1) Complexity, which is inverse to practicality, and hence deals with
the number of notes as which is related to the commas that are being
tempered out
2) Some concept of "error," which could be how the primes or "key
ratios" are mistuned.
3) Your idea to use the HE curve for the above is a mixture of 1) and
2). Using Sethares' curve might well be a more accurate way to do it
as well if you prioritize timbral fusion over the harmonicity of the
fundamentals. You can get used to the lack of timbral fusion and focus
more on fundamental harmonicity, which is why everyone can still hear
a 12-tet major triad as consonant, I think.
4) Perhaps you could delve into the MOS's that are supported by each
one - that is, 12-tet has relatively few MOS's when compared to
something like 17-tet, which has way more since every generator is
coprime with 17.

Just some random ideas I threw out there... :) What I'm really saying
is that I would find such a measure useful, so I encourage you to keep
developing it. If you don't, I'll probably delve into it myself at
some point.

What did you think about Gene's logflat badness, btw? Have you ever
played with that before?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> It works out as a reliable guide to a certain type of concordance. I
> have this feeling the curve would look very, very different if it were
> calculated with harmonic timbres instead of sine waves. Small
> mistunings would produce huge increases in entropy, I think, since
> you'd start seeing all kinds of 30-50 cent dyads popping up in the
> upper partials.

Yeah, that's kind of how I imagine it, too--it'd be a much "lumpier" curve. Trouble is, it would also look very different depending on amplitude of partials...i.e. with the fuzz pedal on vs. with the fuzz pedal off. Well, the minima will stay minima, I'd think, but you'd get a whole lot more maxima.

> Nonsense! All you need is some kind of "badness" measure. Something
> that takes into account the following factors:
> 1) Complexity, which is inverse to practicality, and hence deals with
> the number of notes as which is related to the commas that are being
> tempered out
> 2) Some concept of "error," which could be how the primes or "key
> ratios" are mistuned.

So far it sounds like you're saying "look at each EDO as a manifestation of a regular temperament". I mean, I have to know which intervals to call the "primes", and while that's common sense for many EDOs, when you start getting into the 30's (or even some of the 20's) it's a little more confusing. At 35-EDO, you have two potential 3rd Harmonics that are pretty close in error.

> 3) Your idea to use the HE curve for the above is a mixture of 1) and
> 2). Using Sethares' curve might well be a more accurate way to do it
> as well if you prioritize timbral fusion over the harmonicity of the
> fundamentals. You can get used to the lack of timbral fusion and focus
> more on fundamental harmonicity, which is why everyone can still hear
> a 12-tet major triad as consonant, I think.

Maybe, but I still think that neither the H.E. curve nor the Sethares curve really describes the experience of hearing these intervals. Sethares' curve does actually seem like a better fit, the more I stare at it, but I just don't know that it's going to produce values that are any more helpful than H.E....which is to say, I don't know that it's going to be any more helpful than me saying "these triads sound really stable, but these other triads don't, in this tuning".

> 4) Perhaps you could delve into the MOS's that are supported by each
> one - that is, 12-tet has relatively few MOS's when compared to
> something like 17-tet, which has way more since every generator is
> coprime with 17.

Well, yes, I am definitely doing this, and have planned to all along. But that's a different area of comparison all together.

> Just some random ideas I threw out there... :) What I'm really saying
> is that I would find such a measure useful, so I encourage you to keep
> developing it. If you don't, I'll probably delve into it myself at
> some point.

If you do, I'll be curious as to the results, but looking at my currently available options, I'm not sure there *is* a truly useful measure that won't come with several "caveats" from me (i.e. how, with H.E. as my model, I have to say "just because this pure Major 3rd has a lower H.E. value than this mistuned Perfect 5th, it doesn't mean the mistuned fifth sounds more stable"). Unless I can find a better model, I don't think I have a choice but to abandon these quantitative approaches. Though maybe it will be useful to point out intervals that come close (i.e. 7 cents or so) to ratios with low Tenney Height...I dunno, I need to meditate on it some more.

> What did you think about Gene's logflat badness, btw? Have you ever
> played with that before?

I didn't understand the concept.

-Igs

Hi Igs!

> Sorry, I don't really get what you're doing here.

I think the fact that it's pairwise may still run into the
same problems regarding subharmonic chords. I only have to
construct a subharmonic chord among the partials of two
tones to do raise this doubt, so that's what I did.

> But that's okay, because the more I think about it, the more
> I think this idea isn't really worth pursuing. I mean, I'm
> looking for some quantifiable way to easily compare the
> concordance of EDOs along a single dimension. I thought
> H.E. was the way to go for a while, but now I'm not entirely
> sure. I mean, it works great for sine waves, but for harmonic
> timbres the interactions between partials will play a stronger
> role in determining concordance (or "sensory consonance" if
> that's a better term).

"Sensory consonance" is what people doing psychoacoustics
used for concordance. Except since almost everybody was
working exclusively on roughness it's sometimes associated
with roughness in particular. "Concordance" is better because
it has no such association. It's also better because it's a
single word. If you want just roughness say "roughness".

> I mean, with a sawtooth wave or a distorted guitar, there's no
> way 400 cents is going to sound smoother/more relaxed than
> 320 cents,

Tried it? I tend to doubt it, but I'm willing to listen
to examples.

> 720 cents has lower H.E. than both, so I just don't think
> H.E. is going to work out as a reliable guide to concordance,
> even though I would think it should.

This I believe is more of a problem, though again audio
examples are always good. There are two things worth noting:

1. Sethares' dissonance curve seems to have the same problem!
Zoom in on fig. 3 here:
http://eceserv0.ece.wisc.edu/~sethares/paperspdf/consonance.pdf

2. Vos-curve harmonic entropy should ameliorate the problem
somewhat. Have a look at:
/tuning/files/dyadic/secor4.gif

> Perhaps I should give up on the idea that different EDOs
> can be quantitatively compared along a single dimension,
> or that it's even worthwhile to quantify qualitative
> properties of sound? Maybe I can get the point across
> about what each EDO is and isn't good for just by words.

Did you see this message:

/tuning/topicId_92350.html#92361

?

-Carl

On Tue, Sep 7, 2010 at 5:01 PM, cityoftheasleep <igliashon@...> wrote:
>
> So far it sounds like you're saying "look at each EDO as a manifestation of a regular temperament". I mean, I have to know which intervals to call the "primes", and while that's common sense for many EDOs, when you start getting into the 30's (or even some of the 20's) it's a little more confusing. At 35-EDO, you have two potential 3rd Harmonics that are pretty close in error.

Regular temperament is just the best way of looking at music that I
have. If you have a better way, try that :)

But you could calculate it up to the n-limit. As in, what's 12-tet's
"goodness" value for the 5-limit, the 7-limit, etc?

What would be ideal is to see what naturally emerges - e.g. 7-tet is
great as a 13-limit tuning with no 5's. There is no point judging how
well it approximates ratios of 5 and counting that against it when
they aren't there.

> Maybe, but I still think that neither the H.E. curve nor the Sethares curve really describes the experience of hearing these intervals. Sethares' curve does actually seem like a better fit, the more I stare at it, but I just don't know that it's going to produce values that are any more helpful than H.E....which is to say, I don't know that it's going to be any more helpful than me saying "these triads sound really stable, but these other triads don't, in this tuning".

So why not combine them?

> If you do, I'll be curious as to the results, but looking at my currently available options, I'm not sure there *is* a truly useful measure that won't come with several "caveats" from me (i.e. how, with H.E. as my model, I have to say "just because this pure Major 3rd has a lower H.E. value than this mistuned Perfect 5th, it doesn't mean the mistuned fifth sounds more stable"). Unless I can find a better model, I don't think I have a choice but to abandon these quantitative approaches. Though maybe it will be useful to point out intervals that come close (i.e. 7 cents or so) to ratios with low Tenney Height...I dunno, I need to meditate on it some more.

Why not just incorporate both of them? The entropy is important for
coming up with consonant sounding chords, and the roughness measure is
important for making sure they don't sound ridiculous when played with
actual instruments.

It is also worth mentioning that as we've seen, chords played with
sines often sound great even if the dyads are of high entropy. These
chords can be perceived as fitting more than one fundamental (as can
the minor triad). So HE might not be such a good example of "potential
musicalness" or something like that. Neither, I think, is Sethares'
curve, but some combination of the two might be more useful for that.

-Mike

Igs> Perhaps I should give up on the idea that different EDOs
> can be quantitatively compared along a single dimension,
> or that it's even worthwhile to quantify qualitative
> properties of sound?

I think they can be compared...but not along a single dimension (or at least
without some outright huge exceptions here and there).
People may hate me for this, but I'll say the following:
A) Harmonic Entropy, Roughness, "rooted-ness", Tenney Height, Odd-Limit, or any
other mathematical method in existence (so far as I know) all have at least a
few blatant failures to match what listeners actually judge (on the average) as
consonance.
B) Picking as many inferences as possible from existing theories that get near
what you hear as consonance (IE pick the same ratios and chords as preferable)
seems to be a good basis.
But what about things like explaining the consonance (to many people,
apparently NOT just me) of 15/8 vs. the lower limit nearby 9/5 and 11/6 or
40/27's being less consonant than 22/15 despite HE and critical band consonance
saying otherwise OR (what we agreed on before) that 22/15 is more consonant
sounding than the lower limit 16/11 despite virtually every theory around saying
otherwise?
For those, I'd argue both having your own ears say there is a clear
difference and having others on this list (say 3 or some people) back you up
should be enough for you to say such intervals work. EVEN without a long
mathematical backbone for them. After all...your end audience is more than
musician than the mathematician and ears ultimately are a more powerful judge
than numbers, right?

A side note: Plomp and Llevelt's curve (behind Sethares' work) was based on
simple observations of listeners IE litmus tests of "how consonant do you THINK
this dyad sounds?"...and yet the results in many ways are similar to the very
mathematically solid (IE from a single equation) Harmonic Entropy curve. It
seems proof that any sort of suspicion that having a single equation summarize
everything is nearly always a superior way to rate consonance is simply not
true.

MikeB>"So why not combine them?" (them as in critical band and Harmonic Entropy
theory)
As I've said so many times (agreed!). :-D And I'm saying it again. And Carl
is probably going to blast me yet again for saying that's wrong (as he has
virtually every time before)...but oh well.

Again (regardless of what I read of these theories or am "forced to agree
with to prove understanding of them"), my ear seems to say Sethares curve works
better for small intervals (IE it shows that around 13/12 down to 18/17 or so
the dissonance starts growing exponentially more than it does between 9/8 and
12/11) and HE seems to do a better job about capturing the idea that, for
example, anything around 4/3 and 5/4 sounds much like those intervals and, for
example, that something like 33/25 (1.32) really isn't going to differ in
possible musical purpose from 4/3 (IE 4/3). Again some may whine about "ok what
formula do you have to prove these?" to which I reply "I try to find formulas
that fit what I hear...and if it's my ears vs. math saying something utterly
different, sorry, but I'm not forcing my ears to chase math (and that decision
does not mean I don't understand what the math/formulas are trying to say is
always true)!" If you have lack of faith in anything I say, at least try my
examples with your own ears first...

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> But what about things like explaining the consonance (to many people,
> apparently NOT just me) of 15/8 vs. the lower limit nearby 9/5 and 11/6 or

Not to me. I think it would help if we had some actual test data, because you keep making assertions about what you think as if they will probably be what other people think, and yet often what you claim doesn't make sense to me.

On Tue, Sep 7, 2010 at 11:18 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> > But what about things like explaining the consonance (to many people,
> > apparently NOT just me) of 15/8 vs. the lower limit nearby 9/5 and 11/6 or
>
> Not to me. I think it would help if we had some actual test data, because you keep making assertions about what you think as if they will probably be what other people think, and yet often what you claim doesn't make sense to me.

You say that 11/6 sounds more consonant than 15/8? In what sense do
you mean "consonant?"

-Mike

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

> You say that 11/6 sounds more consonant than 15/8? In what sense do
> you mean "consonant?"

What I really objected to was you claim that 15/8 is more consonant than 9/5. But 11/6 is less harsh also.

On Tue, Sep 7, 2010 at 11:38 PM, genewardsmith
<genewardsmith@...> wrote:
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > You say that 11/6 sounds more consonant than 15/8? In what sense do
> > you mean "consonant?"
>
> What I really objected to was you claim that 15/8 is more consonant than 9/5. But 11/6 is less harsh also.

I don't think I ever said that it was more consonant than 9/5. Michael
Sheiman might have said that. I said that I find major 7 chords
consonant in a way that 4:5:6:7:9:11 is not, and I think it has to do
with some kind of internalized diatonic map that I have.

If I play just 8:10:15, it still sounds like a maj7 chord. Dropping
the 5th out of the chord is one of the oldest tricks in the book. And
when I play just a bare maj7, it sounds like the simplest possible
thing that "is" a major 7 chord. That is, unless I put out some kind
of mental effort to reframe that dyad as part of a minor/^7 chord or
an aug^7 chord or something.

This is taught in most jazz schools under the moniker "shell
voicings," so it's hardly a phenomenon restricted to my ears alone.

-Mike

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

> This is taught in most jazz schools under the moniker "shell
> voicings," so it's hardly a phenomenon restricted to my ears alone.

Give me a break. They don't teach about chords with 11/6 intervals in jazz school, so it's hardly relevant to the comparison.

On Wed, Sep 8, 2010 at 12:14 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > This is taught in most jazz schools under the moniker "shell
> > voicings," so it's hardly a phenomenon restricted to my ears alone.
>
> Give me a break. They don't teach about chords with 11/6 intervals in jazz school, so it's hardly relevant to the comparison.

LOL, what about the rest of my post? You took the part that was meant
to illuminate that lots of people find major 7ths to be consonant and
in no need of resolution, and applied it out of context to the part
that compared 11/6 and 15/8.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Wed, Sep 8, 2010 at 12:14 AM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > This is taught in most jazz schools under the moniker "shell
> > > voicings," so it's hardly a phenomenon restricted to my ears alone.
> >
> > Give me a break. They don't teach about chords with 11/6 intervals in jazz school, so it's hardly relevant to the comparison.
>
> LOL, what about the rest of my post? You took the part that was meant
> to illuminate that lots of people find major 7ths to be consonant and
> in no need of resolution, and applied it out of context to the part
> that compared 11/6 and 15/8.

LOL yourself. Your comment was irrelevant to the point at issue.

Mike wrote:

> > Give me a break. They don't teach about chords with 11/6
> > intervals in jazz school, so it's hardly relevant to the
> > comparison.
>
> LOL, what about the rest of my post? You took the part that was meant
> to illuminate that lots of people find major 7ths to be consonant and
> in no need of resolution, and applied it out of context to the part
> that compared 11/6 and 15/8.

Just jumping in here, but I failed to see how your comments
about triads and tetrads had to do with Gene's about (apparently)
dyads. -C.

On Wed, Sep 8, 2010 at 12:23 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Wed, Sep 8, 2010 at 12:14 AM, genewardsmith
> > <genewardsmith@...> wrote:
> > >
> > > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > > This is taught in most jazz schools under the moniker "shell
> > > > voicings," so it's hardly a phenomenon restricted to my ears alone.
> > >
> > > Give me a break. They don't teach about chords with 11/6 intervals in jazz school, so it's hardly relevant to the comparison.
> >
> > LOL, what about the rest of my post? You took the part that was meant
> > to illuminate that lots of people find major 7ths to be consonant and
> > in no need of resolution, and applied it out of context to the part
> > that compared 11/6 and 15/8.
>
> LOL yourself.

No you!

> Your comment was irrelevant to the point at issue.

The real point at issue, at least my issue, is that I come from a
different musical background than most people on this list. I have
often heard both the major 7 chord and the maj7 dyad as being cited as
"having to resolve" upward, and seen lots of theories about why this
is. But the point is that I -don't- hear it as having to resolve, and
I hear it as a stable sonority. Thus I am very sensitive to musical or
psychoacoustic theories that emerge from common practice trends, and
reflect nothing fundamental about the auditory system or the universe
per se.

Every time I say this, you start mentioning that this is "just me," as
if you want to frame the situation as me being in some kind of
outlying minority not representative of most of the world. I would say
that if we're going to take that approach, then most of America would
say that 15/8 is way more consonant than 11/6, since they don't have a
clue what 11/6 is. They'd probably find it also more consonant than
13/8, and 11/9, and other dyads of lower Tenney height and odd-limit
as well. And major 7 chords as well as major 7 dyadic "shell voicings"
have been a part of popular music for about the last century now and
featured as consonances.

But I would rather not take that approach at all. I'd rather leave it
at there being some kind of subjective component to "consonance."
Would you not agree? And as someone who has worked with microtonal
intervals and modern "jazz" harmony or whatever you'd like to call it,
I am someone who hears both 15/8 and 11/6 as consonant, but in two
different ways. I also hear 4:5:6:7 and 8:10:12:15 consonant in two
different ways.

-Mike

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

> The real point at issue, at least my issue, is that I come from a
> different musical background than most people on this list. I have
> often heard both the major 7 chord and the maj7 dyad as being cited as
> "having to resolve" upward, and seen lots of theories about why this
> is.

You think that has something to do with whether 11/6 or 15/8 is more consonant? Please!

> Every time I say this, you start mentioning that this is "just me,"

No, I object to your constant claims about which dyads are more consonant. based on your own often idiosyncratic reactions.

as
> if you want to frame the situation as me being in some kind of
> outlying minority not representative of most of the world. I would say
> that if we're going to take that approach, then most of America would
> say that 15/8 is way more consonant than 11/6, since they don't have a
> clue what 11/6 is.

I suggest a valid hearing test would be cross-cultural as much as possible.

On Wed, Sep 8, 2010 at 12:40 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The real point at issue, at least my issue, is that I come from a
> > different musical background than most people on this list. I have
> > often heard both the major 7 chord and the maj7 dyad as being cited as
> > "having to resolve" upward, and seen lots of theories about why this
> > is.
>
> You think that has something to do with whether 11/6 or 15/8 is more consonant? Please!

That's quite the strawman. In my initial post where I described my
impression of the maj7 chord, I distinguished between a lot of
auditory phenomena lumped together under the "consonance" moniker.
This is a fact that you seem to be ignoring.

> > Every time I say this, you start mentioning that this is "just me,"
>
> No, I object to your constant claims about which dyads are more consonant. based on your own often idiosyncratic reactions.

...What constant claims? My "constant claims?" Are you confusing me
with the other Mike?

> as
> > if you want to frame the situation as me being in some kind of
> > outlying minority not representative of most of the world. I would say
> > that if we're going to take that approach, then most of America would
> > say that 15/8 is way more consonant than 11/6, since they don't have a
> > clue what 11/6 is.
>
> I suggest a valid hearing test would be cross-cultural as much as possible.

I suggest that there is a definition of the word "consonance" for
which your statement holds true, and another one in which it doesn't.
If you're talking about consonance as in how much an interval fuses
into a single note, that's rooted in psychoacoustics, and I don't
think that really translates directly over to pleasantness, especially
when placed in the context of a chord.

-Mike

On Wed, Sep 8, 2010 at 12:25 AM, Carl Lumma <carl@...> wrote:
>
> Just jumping in here, but I failed to see how your comments
> about triads and tetrads had to do with Gene's about (apparently)
> dyads. -C.

They're related in that the relationship between 4:5:6:7 and
8:10:12:15 is analogous to the relationship between 11:6 and 15:8.
4:5:6:7 is more consonant in the sense that it fuses better, but I
also find 8:10:12:15 consonant in a different way in which 4:5:6:7 is
not. 11:6 fuses better than 15:8, but I find 15:8 to be consonant in a
way that 11:6 is not. 4:5:6:7:9:11 fuses better than
16:20:24:30:36:45, but I find 16:20:24:30:36:45 to be consonant in a
way that 4:5:6:7:9:11 is not.

If we're defining consonance to mean timbral fusion, in the
psychoacoustic sense, then the former chord in each example is more
"consonant." If we're defining it in the more vague sense of whether
an interval chord "sounds pleasant," then both chords are pleasant in
different ways and equally so. I wouldn't say that the first
definition translates directly over to the second one.

I would also say that the statement "8:15 is dissonant" is not a true
statement by the second definition. At least not in my world.

-Mike

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

> ...What constant claims? My "constant claims?" Are you confusing me
> with the other Mike?

Come to think of it, probably. But I get tired of hearing
what are apparently personal preferences elevated to a status I don't see they deserve, especially if I'm not even allowed to call sqrt(2) "boring" without being jumped on, and that was plainly labeled as a personal reaction by the adjectives I used. Just because something is used in jazz theory does not make it a universal law; I react to that the way you do to claims about how thing "should" work based on 18th and 19th century classical music.

And the bottom line is, 11/6 has nothing to do with jazz theory, so trying to analyze it in that way makes no more sense than trying to analyze it in terms of Mozart's style would.

> If you're talking about consonance as in how much an interval fuses
> into a single note, that's rooted in psychoacoustics, and I don't
> think that really translates directly over to pleasantness, especially
> when placed in the context of a chord.

I think "pleasantness" is too subjective to use as a basis for theory, but am ready to be proven wrong. I suspect one problem is that it is more subject to cultural bias arising from previous exposure.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Wed, Sep 8, 2010 at 12:25 AM, Carl Lumma <carl@...> wrote:
> >
> > Just jumping in here, but I failed to see how your comments
> > about triads and tetrads had to do with Gene's about (apparently)
> > dyads. -C.
>
> They're related in that the relationship between 4:5:6:7 and
> 8:10:12:15 is analogous to the relationship between 11:6 and 15:8.

??????????????

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> I think the fact that it's pairwise may still run into the
> same problems regarding subharmonic chords. I only have to
> construct a subharmonic chord among the partials of two
> tones to do raise this doubt, so that's what I did.

Oh, I think I see what you're saying now.

> > I mean, with a sawtooth wave or a distorted guitar, there's no
> > way 400 cents is going to sound smoother/more relaxed than
> > 320 cents,
>
> Tried it? I tend to doubt it, but I'm willing to listen
> to examples.

I could cobble some together, but I've played in 15-EDO often enough to know that those minor 3rds really "hum" compared to the major 3rds. They're practically JI.

I said:
> > 720 cents has lower H.E. than both, so I just don't think
> > H.E. is going to work out as a reliable guide to concordance,
> > even though I would think it should.

Carl said:
> This I believe is more of a problem, though again audio
> examples are always good. There are two things worth noting:
>
> 1. Sethares' dissonance curve seems to have the same problem!
> Zoom in on fig. 3 here:
> http://eceserv0.ece.wisc.edu/~sethares/paperspdf/consonance.pdf

Yeah, that graph is hard to read. Really I'd want to find a way to put on a "0 to 100" scale the same way I did with H.E.; that's the only way I can *really* compare them.

> 2. Vos-curve harmonic entropy should ameliorate the problem
> somewhat. Have a look at:
> /tuning/files/dyadic/secor4.gif

Dag nabbit, how many H.E. curves are there?? Every time I come up with an issue with one, you pop up with another version that looks a lot better. But still, I think I will find the same issues with clashing partials. Even on that version, it looks like a rather far-off 3/2 will still beat a good 6/5. Not by as much as in the exponential version that I've been using, but still.

Mike mentioned an octave-equivalent version of H.E. I'm curious how that works, because I frankly am starting to have issues with Tenney Height. I don't think it's measuring the same thing I want to measure, because I have a hard time hearing 12/5 as being half as "restful" as 6/5, or hearing 5/2 as twice as "restful" as 5/4. You once described concordance to me as "instantaneously producing a feeling of restfulness" or something, but I don't think that's quite what Tenney Height is measuring. It just boggles me that some intervals go up geometrically in Tenney Height as you increase the octave, while others drop geometrically. That just doesn't fit with my experience of concordance. It also totally blows chunks all over the "critical bandwidth" theory, right? So I'm wondering if the octave-equivalent formulation of H.E. solves that somehow.

> > Perhaps I should give up on the idea that different EDOs
> > can be quantitatively compared along a single dimension,
> > or that it's even worthwhile to quantify qualitative
> > properties of sound? Maybe I can get the point across
> > about what each EDO is and isn't good for just by words.
>
> Did you see this message:
>
> /tuning/topicId_92350.html#92361

Yes, I thought I replied to it too, but I guess Yahoo ate my response. Just FYI, the "re-scaling" I used was the one suggested by S.J. Martin in another response on that thread, basically the same as the exponential but adjusted to be 0-100, with 100 being "lowest H.E." and 0 being highest.

As to the list of 7-limit triads, my reckoning is that 31-EDO would dominate all the triads that include 5, and 36-EDO would dominate all the ones that don't. But I've been thinking about higher odd-limits as well, as a way to incorporate triads formed by combinations of 7-limit dyads. Not necessarily to rank or quantify EDOs, but to help guide people into finding useful chords. I do feel that it's helpful, for instance, to know that in 23-EDO, the chord formed by degrees 0-4-9 is within a 7-cent error of 16:18:21. I mean, 16:18:21 may not be a great chord, but for tunings like 23-EDO, it's one of the better chords going (there's also a pretty good 5:6:7 chord, also, but we all know how I feel about that chord vs. 16:18:21).

I dunno, there are so many different possibilities as to how I can approach this, but they all have their issues and inconsistencies. At least if I leave JI and concordance out of it, and just stick to subjective impressions, I won't have to add caveats every chapter of "the numbers say this, but my experience says otherwise". I've yet to see a formulation of anything that doesn't run afoul of my experience in some place or another.

-Igs

2010/9/8 genewardsmith <genewardsmith@...>:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
>> ...What constant claims? My "constant claims?" Are you confusing me
>> with the other Mike?
>
> Come to think of it, probably.

Haha, ok.

> But I get tired of hearing
> what are apparently personal preferences elevated to a status I don't see they deserve, especially if I'm not even allowed to call sqrt(2) "boring" without being jumped on, and that was plainly labeled as a personal reaction by the adjectives I used.

You're allowed to call sqrt(2) boring as much as you want. It clearly
is a personal reaction. I said that I think there are multiple
psychoacoustic percepts that are often lumped together under the
"consonance" moniker:

1) How much the partials of a sonority clash with one another
2) How much they fuse into a single note
3) How "pleasant" something ends up sounding without regard to any
specific psychoacoustic

I also threw out that there seems to be some other basis for
consonance that has to do with internalized maps, so that someone with
an internalized map for expanded diatonic hearing or whatever you'd
like to call it will hear a major 7 chord as consonant. And I don't
know how it works, but it clearly works somehow, and that in that
sense I find a major 7 chord really consonant, whether the thirds are
intoned 5/4 or 9/7.

Is that not fair?

> Just because something is used in jazz theory does not make it a universal law; I react to that the way you do to claims about how thing "should" work based on 18th and 19th century classical music.
>
> And the bottom line is, 11/6 has nothing to do with jazz theory, so trying to analyze it in that way makes no more sense than trying to analyze it in terms of Mozart's style would.

I don't think that maj7 chords have anything to do with jazz theory
either. I think they have to do with an expanded approach to meantone
that started around the beginning of the 21st century. And I think
that for me to say that they're consonant in the sense that I perceive
how they fit into this very beautiful complex structure (which is an
superset of the same structure that leads to common practice harmony)
is similar to me saying that I see a dalmatian in this picture:

http://www.mindmaptutor.com/wp-content/uploads/2009/08/emergence.jpg

Is it not worth admitting that?

>> If you're talking about consonance as in how much an interval fuses
>> into a single note, that's rooted in psychoacoustics, and I don't
>> think that really translates directly over to pleasantness, especially
>> when placed in the context of a chord.
>
> I think "pleasantness" is too subjective to use as a basis for theory, but am ready to be proven wrong. I suspect one problem is that it is more subject to cultural bias arising from previous exposure.

Then I think that "consonance" is too subjective to reduce to a single
psychoacoustic percept. And I also think that harmonic concordance is
too far removed from consonance to use as a basis for theory, as
people can get used to mavila and hear all sorts of consonances in it.

My current obsession is to try and find some kind of pattern in how
this type of consonance works. It has something to do, I think, with
Rothenberg's idea that people build up mental reference frames or maps
from scales that they've learned. The subjective component to this
would come from what maps people have learned so far and which one
they haven't, so if we assume people have "learned" how to place
things in a diatonic/5+2 sense, we can make predictions on what they'd
find "consonant." I also think that Paul had some more of the puzzle
figured out with his 22 paper, and maybe some other author I haven't
read yet has it all figured out. But either way there is still some
type of consonance emerging from "that," and that's what I'd love to
have figured out. And what's wrong with that?

-Mike

On Wed, Sep 8, 2010 at 1:13 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Wed, Sep 8, 2010 at 12:25 AM, Carl Lumma <carl@...> wrote:
> > >
> > > Just jumping in here, but I failed to see how your comments
> > > about triads and tetrads had to do with Gene's about (apparently)
> > > dyads. -C.
> >
> > They're related in that the relationship between 4:5:6:7 and
> > 8:10:12:15 is analogous to the relationship between 11:6 and 15:8.
>
> ??????????????

I explained exactly how it's analogous right after that sentence.
4:5:6:7 is less complex and fuses better, and 8:10:12:15 has some
consonant quality that emerges from the diatonic map that I have. And
the latter quality emerges even if the thirds aren't intoned
differently. Likewise, 11:6 is less complex and fuses better than
15:8, but the latter draws from the same diatonically consonant
quality that makes 8:10:12:15 so consonant.

-Mike

Hi Igs,

> > 1. Sethares' dissonance curve seems to have the same problem!
> > Zoom in on fig. 3 here:
> > http://eceserv0.ece.wisc.edu/~sethares/paperspdf/consonance.pdf
>
> Yeah, that graph is hard to read. Really I'd want to find a
> way to put on a "0 to 100" scale the same way I did with H.E.;
> that's the only way I can *really* compare them.

Yes, that'd be nice. But it's evident that it has the same
problem. The tick marks are every 100 cents, so 720 is about
a fifth of the way across.

> > 2. Vos-curve harmonic entropy should ameliorate the problem
> > somewhat. Have a look at:
> > /tuning/files/dyadic/
> > secor4.gif
>
> Dag nabbit, how many H.E. curves are there??

Paul did a lot of work with deep and experienced listeners
like George Secor, Margo Shulter, and many many others as he
was experimenting with harmonic entropy.
The Vos curve version is based on the work of Joos Vos, who
did experiments in which listeners rated the purity of
mistuned intervals. I don't have the paper that Paul had
so I can't comment further, but I believe Vos found that
purity ratings obeyed an exponential distribution
http://en.wikipedia.org/wiki/Exponential_distribution
around just ratios. (Please note that this is NOT the same
as taking the exponential of entropy at the end, as you've
been doing.)
Regular h.e. uses a Gaussian distribution instead, which is
the most natural assumption and the one used by Goldstein in
his famous 'central pitch processor' model.
Those are the two principle versions. Beyond that, it's
mainly how much noise you want to assume (the variable "s")
and whether you want to take the exponential when you're
done. Goldstein deduced that s was about 1%, but Paul tried
values between 0.6% and 1.2%. Making it smaller causes more
local minima to appear.
Taking the exponential at the end is perfectly OK since it
doesn't change the ranking of intervals.

> Mike mentioned an octave-equivalent version of H.E. I'm
> curious how that works,

I think Paul just averaged the entropy of every pair of
inversions. Like 16/15 and 15/8 would always be the same.
So generally it will lead to worse results. It's only
useful if you insist on using octave equivalence in your
music theory.

> I have a hard time hearing 12/5 as being half as "restful"
> as 6/5, or hearing 5/2 as twice as "restful" as 5/4.

In both cases it's not twice, but root twice. I definitely
hear 5/2 as sweeter than 5/4. I was dubious about 6/5 vs
12/5 but you know, I just tried it in Scala in two different
octaves and I have to say, to my surprise 6/5 sounds more
concordant.

> It just boggles me that some intervals go up geometrically
> in Tenney Height as you increase the octave, while others
> drop geometrically. That just doesn't fit with my experience
> of concordance.

To make 5/2 in a harmonic series, you use lower harmonics
than to make 5/4. And same with 6/5 and 12/5.

> It also totally blows chunks all over the "critical bandwidth"
> theory, right?

I believe that making a dyad wider by an octave always
decreases roughness according to Sethares' model, but don't
quote me on that.

> So I'm wondering if the octave-equivalent
> formulation of H.E. solves that somehow.

I don't think so. However you may be on the right track
because your observations seem to be calling for SOME
discount on factors of 2.

By the way, you should know that I almost can't believe I'm
having this conversation with you. I was driving -- I think
it was the week before I left for burning man -- playing my
iPhone on shuffle through my car, and Numerology came up.
And I was totally floored. What insanely incredible music!
I mean, like, you're a genius. Way better at music than I
am at music theory. Then I had the same f'ing experience
today when listening to Aaron Johnson's Peer Gynt.
I mean, damn.

> > Did you see this message:
> > /tuning/topicId_92350.html#92361
>
> Yes, I thought I replied to it too, but I guess Yahoo ate my
> response. Just FYI, the "re-scaling" I used was the one
> suggested by S.J. Martin in another response on that thread,
> basically the same as the exponential but adjusted to be 0-100,
> with 100 being "lowest H.E." and 0 being highest.

Gotcha.

> As to the list of 7-limit triads, my reckoning is that 31-EDO
> would dominate all the triads that include 5, and 36-EDO would
> dominate all the ones that don't.

Ah, but that's why multiply the error by EDO size -- getting
badness!

> At least if I leave JI and concordance out of it, and just
> stick to subjective impressions, I won't have to add caveats
> every chapter of "the numbers say this, but my experience
> says otherwise".

I think it would be valuable either way.

-Carl

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

> > > They're related in that the relationship between 4:5:6:7 and
> > > 8:10:12:15 is analogous to the relationship between 11:6 and
> > > 15:8.
> >
> > ??????????????
>
> I explained exactly how it's analogous right after that sentence.

Not in a way that made sense to me...

> 4:5:6:7 is less complex and fuses better, and 8:10:12:15 has some
> consonant quality that emerges from the diatonic map that I have.

8:10:12:15 is concordant even without a map. But that's
also got NOTHING TO DO with 15:8 vs 11:6 or any other
comparison of dyads. It may REMIND you of it in some vague
way that nobody else has a chance of understanding, but
that's about it.

> Likewise, 11:6 is less complex and fuses better than
> 15:8,

Does it? The two are only marginally distinct. Neither
ratio is particularly tunable by ear.

-Carl

This posting is copyright Charles Lucy 2010, and explains how this consonance/dissonance conundrum actually works for Western harmony.

Forget about your addictive integer frequency ratios. They are all red herrings.

If you want to understand Western harmony and jazz you need to use a meantone-type model

Here follows the solution:

A theory for the ranking of consonance and dissonance between simultaneously sounding notes.

Assumptions:
a) Notes played simultaneously, which are closer on the spiral of fourths and fifths, regardless of ocatve, tend to sound more consonant, than those which are seperated by a greater number of steps of fourths and fifths.
b) An "Index of Dissonance" may be calculated by adding the number of steps between each note and every other simultaneously sounding note, and dividing the result by the number of intervals which have been used in the addition.
c) All notes are LucyTuned; i.e. meantone intervals (5 Large + 2 small per octave (ratio 2))
d) Large interval = 1200 cents/(2*pi) = 190.9858 cents or ratio of 1.116633.

Method:
a) List the notes to be considered.
b) Count the number of steps of fourths or fifths for each pair of notes to be played simulanteously.
c) Add the resulting steps for each pair.
d) Divide the resulting sum by the number of pairs considered.
e) The Index of Dissonance will be constant regardless of key or notenames, and may be considered as an index of dissonance for a type of chord or for a scale.

Examples:

(Using the spiral pattern from fourths to fifths of Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B# for first 22 notes.)

e.g. 1:

Consider the notes: C-E-G-B (Chord is C Major Seventh)

Between C and E = 4 steps
Between C and G = 1 step
Between C and B = 5 steps
Between E and G = 3 steps
Between E and B = 1 step
Between G and B = 4 steps

Total number of steps = 18 steps

Number of intervals considered = 6 intervals

Index of Dissonance = 18/6 = 3.0

e.g. 2:

Consider the notes: C-E-G# (Chord is C Augmented Triad)

Between C and E = 4 steps
Between C and G# = 8 steps
Between E and G# = 4 steps

Total number of steps = 16 steps

Number of intervals considered = 3 intervals

Index of Dissonance = 16/3 = 5.3333

e.g. 3:

Consider the notes: C-Eb-Gb-Bbb (Chord is C Diminished Seventh)

Between C and Eb = 3 steps
Between C and Gb = 6 step
Between C and Bbb = 9 steps
Between Eb and Gb = 3 steps
Between Eb and Bbb = 6 step
Between Gb and Bbb = 3 steps

Total number of steps = 30 steps

Number of intervals considered = 6 intervals

Index of Dissonance = 30/6 = 5.0

e.g. 4:

Consider the notes: C-Eb-G-B (Chord is C minor Major Seventh)

Between C and Eb = 3 steps
Between C and G = 1 step
Between C and B = 5 steps
Between Eb and G = 4 steps
Between Eb and B = 8 steps
Between G and B = 4 steps

Total number of steps = 25 steps

Number of intervals considered = 6 intervals

Index of Dissonance = 25/6 = 4.16667

e.g. 5:

Consider the notes: C-E-G (Chord is C Major)

Between C and E = 4 steps
Between C and G = 1 step
Between E and G = 3 steps

Total number of steps = 8 steps

Number of intervals considered = 3 intervals

Index of Dissonance = 8/3 = 2.66667

e.g. 6:

Consider the notes: C-Eb-G (Chord is C minor)

Between C and Eb = 3 steps
Between C and G = 1 step
Between Eb and G = 4 steps

Total number of steps = 8 steps

Number of intervals considered = 3 intervals

Index of Dissonance = 8/3 = 2.66667

e.g. 7:

Consider the notes: C-Eb-Gb (Chord is C Diminished Triad)

Between C and Eb = 3 steps
Between C and Gb = 6 step
Between Eb and Gb = 3 steps

Total number of steps = 12 steps

Number of intervals considered = 3 intervals

Index of Dissonance = 12/3 = 4.0

e.g. 8:

Consider the notes: C-E-G-Bb (Chord is C Seventh)

Between C and E = 4 steps
Between C and G = 1 step
Between C and Bb = 2 steps
Between E and G = 3 steps
Between E and Bb = 6 steps
Between G and Bb = 3 steps

Total number of steps = 19 steps

Number of intervals considered = 6 intervals

Index of Dissonance = 19/6 = 3.166667

e.g. 9:

Consider the notes: C-E-G-A (Chord is C Sixth or A minor Seventh)

Between C and E = 4 steps
Between C and G = 1 step
Between C and A = 3 steps
Between E and G = 3 steps
Between E and A = 1 steps
Between G and A = 2 steps

Total number of steps = 13 steps

Number of intervals considered = 6 intervals

Index of Dissonance = 13/6 = 2.166667

e.g. 10:

Consider the notes: C-E-G-Bb-D) (Chord is C Ninth)

Between C and E = 4 steps
Between C and G = 1 step
Between C and Bb = 2 steps
Between C and D = 2 steps
Between E and G = 3 steps
Between E and Bb = 6 steps
Between E and D = 2 steps
Between G and Bb = 3 steps
Between G and D = 1 step
Between Bb and D = 4 step

Total number of steps = 28 steps

Number of intervals considered = intervals

Index of Dissonance = 28/10 = 2.8

e.g. 11:

Consider the notes: C-E-G-D (Chord is C9 - no 7th)

Between C and E = 4 steps
Between C and G = 1 step
Between C and D = 2 steps
Between E and G = 3 steps
Between E and D = 2 steps
Between G and D = 1 step

Total number of steps = 13 steps

Number of intervals considered = 6 intervals

Index of Dissonance = 13/6 = 2.166667

e.g. 12:

Consider the notes: C-E-G-Bb-A) (Chord is C Thirteen)

Between C and E = 4 steps
Between C and G = 1 step
Between C and Bb = 2 steps
Between C and A = 3 steps
Between E and G = 3 steps
Between E and Bb = 6 steps
Between E and A = 1 step
Between G and Bb = 3 steps
Between G and A = 2 steps
Between Bb and A = 5 steps

Total number of steps = 30 steps

Number of intervals considered = intervals

Index of Dissonance = 30/10 = 3.0

Ranking of Indices of Dissonance:
Sixth or Minor Seventh = 2.1666667 (C6 or Am7)
Ninth No Seventh = 2.166667 (C9 - no 7th)
Major Triad = 2.66667 (C)
Minor Triad = 2.66667 (Cm)
Ninth = 2.8 (C9)
Major Seventh = 3.0 (CMaj7)
Thirteenth = 3.0 (C13)
Dominant Seventh 3.16666667 (C7)
Diminished Triad = 4.0 (Cdim)
Minor Major Seventh = 4.16667 (CmMaj7)
Diminished Seventh = 5 (Cdim7)
Augmented Triad = 5.33333 (CAug)

In this way it is possible to assign a dissonance index to all scales. harmony, and chords.

On 8 Sep 2010, at 06:29, Mike Battaglia wrote:

> 2010/9/8 genewardsmith <genewardsmith@...>:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> >> ...What constant claims? My "constant claims?" Are you confusing me
> >> with the other Mike?
> >
> > Come to think of it, probably.
>
> Haha, ok.
>
> > But I get tired of hearing
> > what are apparently personal preferences elevated to a status I don't see they deserve, especially if I'm not even allowed to call sqrt(2) "boring" without being jumped on, and that was plainly labeled as a personal reaction by the adjectives I used.
>
> You're allowed to call sqrt(2) boring as much as you want. It clearly
> is a personal reaction. I said that I think there are multiple
> psychoacoustic percepts that are often lumped together under the
> "consonance" moniker:
>
> 1) How much the partials of a sonority clash with one another
> 2) How much they fuse into a single note
> 3) How "pleasant" something ends up sounding without regard to any
> specific psychoacoustic
>
> I also threw out that there seems to be some other basis for
> consonance that has to do with internalized maps, so that someone with
> an internalized map for expanded diatonic hearing or whatever you'd
> like to call it will hear a major 7 chord as consonant. And I don't
> know how it works, but it clearly works somehow, and that in that
> sense I find a major 7 chord really consonant, whether the thirds are
> intoned 5/4 or 9/7.
>
> Is that not fair?
>
> > Just because something is used in jazz theory does not make it a universal law; I react to that the way you do to claims about how thing "should" work based on 18th and 19th century classical music.
> >
> > And the bottom line is, 11/6 has nothing to do with jazz theory, so trying to analyze it in that way makes no more sense than trying to analyze it in terms of Mozart's style would.
>
> I don't think that maj7 chords have anything to do with jazz theory
> either. I think they have to do with an expanded approach to meantone
> that started around the beginning of the 21st century. And I think
> that for me to say that they're consonant in the sense that I perceive
> how they fit into this very beautiful complex structure (which is an
> superset of the same structure that leads to common practice harmony)
> is similar to me saying that I see a dalmatian in this picture:
>
> http://www.mindmaptutor.com/wp-content/uploads/2009/08/emergence.jpg
>
> Is it not worth admitting that?
>
> >> If you're talking about consonance as in how much an interval fuses
> >> into a single note, that's rooted in psychoacoustics, and I don't
> >> think that really translates directly over to pleasantness, especially
> >> when placed in the context of a chord.
> >
> > I think "pleasantness" is too subjective to use as a basis for theory, but am ready to be proven wrong. I suspect one problem is that it is more subject to cultural bias arising from previous exposure.
>
> Then I think that "consonance" is too subjective to reduce to a single
> psychoacoustic percept. And I also think that harmonic concordance is
> too far removed from consonance to use as a basis for theory, as
> people can get used to mavila and hear all sorts of consonances in it.
>
> My current obsession is to try and find some kind of pattern in how
> this type of consonance works. It has something to do, I think, with
> Rothenberg's idea that people build up mental reference frames or maps
> from scales that they've learned. The subjective component to this
> would come from what maps people have learned so far and which one
> they haven't, so if we assume people have "learned" how to place
> things in a diatonic/5+2 sense, we can make predictions on what they'd
> find "consonant." I also think that Paul had some more of the puzzle
> figured out with his 22 paper, and maybe some other author I haven't
> read yet has it all figured out. But either way there is still some
> type of consonance emerging from "that," and that's what I'd love to
> have figured out. And what's wrong with that?
>
> -Mike
>

Charles Lucy
lucy@...

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can found at:

http://www.lullabies.co.uk

On Wed, Sep 8, 2010 at 6:18 AM, Charles Lucy <lucy@...> wrote:

This posting is copyright Charles Lucy 2010,

--------------------------

This is a joke, right?

Chris

'Fraid not.

It makes C-E more dissonant than C-D.

It doesn't really address 7 or 11 or 13 ratios. (Guess those don't exist.)

It has fifths that are flat.

It's a meantone system, so it doesn't get you anything new, really. However, it has the virtue of obscurity.

It's numerological.

It gives a false sense of precision about consonance and dissonance.

Other than that, it's just fine.

It's copyrighted, somehow, so I'm afraid to say more.

-c

On Sep 8, 2010, at 8:36 AM, Chris Vaisvil wrote:

> On Wed, Sep 8, 2010 at 6:18 AM, Charles Lucy <lucy@...> wrote:
>
> This posting is copyright Charles Lucy 2010,
>
> --------------------------
>
> This is a joke, right?
>
> Chris
>

Another theory of all?

Just pure hypothesis, and probably wrong. If I understand well, C-D-G is more consonant then C-E-G by your theory, just because E is more far on fifth spiral then D.

Real world is more complex, we must also consider:

- absolute pitch of the chord

- voicing of the chord (narrow, wide)

- inversions

- tuning and temperament

- timbre

- instrumentation

- duration of chord sounding

- absolute volume (amplitude, dynamics) of the chord and its individual notes

- musical context

- cultural context

- knowledge of music and listening experience of listener

- and maybe more...

All this has influence on consonance-dissonance perception.

Besides: more consonant are chords which are part of harmonic series or are near, have similar structure (wider intervals down, narrower intervals up) and have less notes. More dense chords will sound more dissonant, especially when they are build with more "foreign" notes (chromatic or microtuned), and also when they don't follow basic shape of structure of harmonic series (that means they haven't wider intervals down and narrower up). That's obvious, we don't need to confirm this by mathematical equations. It's based on the empirical experience collected through centuries of Western music development.

Just my personal opinion based on my personal experience as an composer and arranger.

IMVHO your solution is too simple to be truth. It's not possible to measure exactly consonancy or dissonancy of chords. Fortunately it's relative, subjective and emotional value. Fortunately for the art. That's the magic of the art, certain unpredictability and resistance to the chains of exact rules. Even so there's a lot of math in the music, it's not necessary to add more :-)

Daniel Forro

On 8 Sep 2010, at 7:18 PM, Charles Lucy wrote:

>
>
> This posting is copyright Charles Lucy 2010, and explains how this > consonance/dissonance conundrum actually works for Western harmony.
>
> Forget about your addictive integer frequency ratios. They are all > red herrings.
>
> If you want to understand Western harmony and jazz you need to use > a meantone-type model
>
> Here follows the solution:
>
> A theory for the ranking of consonance and dissonance between > simultaneously sounding notes.
>
> Assumptions:
> a) Notes played simultaneously, which are closer on the spiral of > fourths and fifths, regardless of ocatve, tend to sound more > consonant, than those which are seperated by a greater number of > steps of fourths and fifths.
> b) An "Index of Dissonance" may be calculated by adding the number > of steps between each note and every other simultaneously sounding > note, and dividing the result by the number of intervals which have > been used in the addition.
> c) All notes are LucyTuned; i.e. meantone intervals (5 Large + 2 > small per octave (ratio 2))
> d) Large interval = 1200 cents/(2*pi) = 190.9858 cents or ratio of > 1.116633.
>
>
> Method:
> a) List the notes to be considered.
> b) Count the number of steps of fourths or fifths for each pair of > notes to be played simulanteously.
> c) Add the resulting steps for each pair.
> d) Divide the resulting sum by the number of pairs considered.
> e) The Index of Dissonance will be constant regardless of key or > notenames, and may be considered as an index of dissonance for a > type of chord or for a scale.
>
>
> Examples:
>
> (Using the spiral pattern from fourths to fifths of Bbb-Fb-Cb-Gb-Db-> Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B# for first 22 notes.)
>
> e.g. 1:
>
> Consider the notes: C-E-G-B (Chord is C Major Seventh)
>
> Between C and E = 4 steps
> Between C and G = 1 step
> Between C and B = 5 steps
> Between E and G = 3 steps
> Between E and B = 1 step
> Between G and B = 4 steps
>
> Total number of steps = 18 steps
>
> Number of intervals considered = 6 intervals
>
> Index of Dissonance = 18/6 = 3.0
>
> e.g. 2:
>
> Consider the notes: C-E-G# (Chord is C Augmented Triad)
>
> Between C and E = 4 steps
> Between C and G# = 8 steps
> Between E and G# = 4 steps
>
>
> Total number of steps = 16 steps
>
> Number of intervals considered = 3 intervals
>
> Index of Dissonance = 16/3 = 5.3333
>
>
> e.g. 3:
>
> Consider the notes: C-Eb-Gb-Bbb (Chord is C Diminished Seventh)
>
> Between C and Eb = 3 steps
> Between C and Gb = 6 step
> Between C and Bbb = 9 steps
> Between Eb and Gb = 3 steps
> Between Eb and Bbb = 6 step
> Between Gb and Bbb = 3 steps
>
> Total number of steps = 30 steps
>
> Number of intervals considered = 6 intervals
>
> Index of Dissonance = 30/6 = 5.0
>
> e.g. 4:
>
> Consider the notes: C-Eb-G-B (Chord is C minor Major Seventh)
>
> Between C and Eb = 3 steps
> Between C and G = 1 step
> Between C and B = 5 steps
> Between Eb and G = 4 steps
> Between Eb and B = 8 steps
> Between G and B = 4 steps
>
> Total number of steps = 25 steps
>
> Number of intervals considered = 6 intervals
>
> Index of Dissonance = 25/6 = 4.16667
>
> e.g. 5:
>
> Consider the notes: C-E-G (Chord is C Major)
>
> Between C and E = 4 steps
> Between C and G = 1 step
> Between E and G = 3 steps
>
> Total number of steps = 8 steps
>
> Number of intervals considered = 3 intervals
>
> Index of Dissonance = 8/3 = 2.66667
>
>
> e.g. 6:
>
> Consider the notes: C-Eb-G (Chord is C minor)
>
> Between C and Eb = 3 steps
> Between C and G = 1 step
> Between Eb and G = 4 steps
>
> Total number of steps = 8 steps
>
> Number of intervals considered = 3 intervals
>
> Index of Dissonance = 8/3 = 2.66667
>
> e.g. 7:
>
> Consider the notes: C-Eb-Gb (Chord is C Diminished Triad)
>
> Between C and Eb = 3 steps
> Between C and Gb = 6 step
> Between Eb and Gb = 3 steps
>
> Total number of steps = 12 steps
>
> Number of intervals considered = 3 intervals
>
> Index of Dissonance = 12/3 = 4.0
>
> e.g. 8:
>
> Consider the notes: C-E-G-Bb (Chord is C Seventh)
>
> Between C and E = 4 steps
> Between C and G = 1 step
> Between C and Bb = 2 steps
> Between E and G = 3 steps
> Between E and Bb = 6 steps
> Between G and Bb = 3 steps
>
> Total number of steps = 19 steps
>
> Number of intervals considered = 6 intervals
>
> Index of Dissonance = 19/6 = 3.166667
>
> e.g. 9:
>
> Consider the notes: C-E-G-A (Chord is C Sixth or A minor Seventh)
>
> Between C and E = 4 steps
> Between C and G = 1 step
> Between C and A = 3 steps
> Between E and G = 3 steps
> Between E and A = 1 steps
> Between G and A = 2 steps
>
> Total number of steps = 13 steps
>
> Number of intervals considered = 6 intervals
>
> Index of Dissonance = 13/6 = 2.166667
>
> e.g. 10:
>
> Consider the notes: C-E-G-Bb-D) (Chord is C Ninth)
>
> Between C and E = 4 steps
> Between C and G = 1 step
> Between C and Bb = 2 steps
> Between C and D = 2 steps
> Between E and G = 3 steps
> Between E and Bb = 6 steps
> Between E and D = 2 steps
> Between G and Bb = 3 steps
> Between G and D = 1 step
> Between Bb and D = 4 step
>
>
> Total number of steps = 28 steps
>
> Number of intervals considered = intervals
>
> Index of Dissonance = 28/10 = 2.8
>
> e.g. 11:
>
> Consider the notes: C-E-G-D (Chord is C9 - no 7th)
>
> Between C and E = 4 steps
> Between C and G = 1 step
> Between C and D = 2 steps
> Between E and G = 3 steps
> Between E and D = 2 steps
> Between G and D = 1 step
>
> Total number of steps = 13 steps
>
> Number of intervals considered = 6 intervals
>
> Index of Dissonance = 13/6 = 2.166667
>
> e.g. 12:
>
> Consider the notes: C-E-G-Bb-A) (Chord is C Thirteen)
>
> Between C and E = 4 steps
> Between C and G = 1 step
> Between C and Bb = 2 steps
> Between C and A = 3 steps
> Between E and G = 3 steps
> Between E and Bb = 6 steps
> Between E and A = 1 step
> Between G and Bb = 3 steps
> Between G and A = 2 steps
> Between Bb and A = 5 steps
>
>
> Total number of steps = 30 steps
>
> Number of intervals considered = intervals
>
> Index of Dissonance = 30/10 = 3.0
>
>
>
> Ranking of Indices of Dissonance:
> Sixth or Minor Seventh = 2.1666667 (C6 or Am7)
> Ninth No Seventh = 2.166667 (C9 - no 7th)
> Major Triad = 2.66667 (C)
> Minor Triad = 2.66667 (Cm)
> Ninth = 2.8 (C9)
> Major Seventh = 3.0 (CMaj7)
> Thirteenth = 3.0 (C13)
> Dominant Seventh 3.16666667 (C7)
> Diminished Triad = 4.0 (Cdim)
> Minor Major Seventh = 4.16667 (CmMaj7)
> Diminished Seventh = 5 (Cdim7)
> Augmented Triad = 5.33333 (CAug)
>
>
> In this way it is possible to assign a dissonance index to all > scales. harmony, and chords.
>
>
> On 8 Sep 2010, at 06:29, Mike Battaglia wrote:
>
>> 2010/9/8 genewardsmith <genewardsmith@...>:
>> >
>> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> >> wrote:
>> >
>> >> ...What constant claims? My "constant claims?" Are you >> confusing me
>> >> with the other Mike?
>> >
>> > Come to think of it, probably.
>>
>> Haha, ok.
>>
>> > But I get tired of hearing
>> > what are apparently personal preferences elevated to a status I >> don't see they deserve, especially if I'm not even allowed to call >> sqrt(2) "boring" without being jumped on, and that was plainly >> labeled as a personal reaction by the adjectives I used.
>>
>> You're allowed to call sqrt(2) boring as much as you want. It clearly
>> is a personal reaction. I said that I think there are multiple
>> psychoacoustic percepts that are often lumped together under the
>> "consonance" moniker:
>>
>> 1) How much the partials of a sonority clash with one another
>> 2) How much they fuse into a single note
>> 3) How "pleasant" something ends up sounding without regard to any
>> specific psychoacoustic
>>
>> I also threw out that there seems to be some other basis for
>> consonance that has to do with internalized maps, so that someone >> with
>> an internalized map for expanded diatonic hearing or whatever you'd
>> like to call it will hear a major 7 chord as consonant. And I don't
>> know how it works, but it clearly works somehow, and that in that
>> sense I find a major 7 chord really consonant, whether the thirds are
>> intoned 5/4 or 9/7.
>>
>> Is that not fair?
>>
>> > Just because something is used in jazz theory does not make it a >> universal law; I react to that the way you do to claims about how >> thing "should" work based on 18th and 19th century classical music.
>> >
>> > And the bottom line is, 11/6 has nothing to do with jazz theory, >> so trying to analyze it in that way makes no more sense than >> trying to analyze it in terms of Mozart's style would.
>>
>> I don't think that maj7 chords have anything to do with jazz theory
>> either. I think they have to do with an expanded approach to meantone
>> that started around the beginning of the 21st century. And I think
>> that for me to say that they're consonant in the sense that I >> perceive
>> how they fit into this very beautiful complex structure (which is an
>> superset of the same structure that leads to common practice harmony)
>> is similar to me saying that I see a dalmatian in this picture:
>>
>> http://www.mindmaptutor.com/wp-content/uploads/2009/08/emergence.jpg
>>
>> Is it not worth admitting that?
>>
>> >> If you're talking about consonance as in how much an interval >> fuses
>> >> into a single note, that's rooted in psychoacoustics, and I don't
>> >> think that really translates directly over to pleasantness, >> especially
>> >> when placed in the context of a chord.
>> >
>> > I think "pleasantness" is too subjective to use as a basis for >> theory, but am ready to be proven wrong. I suspect one problem is >> that it is more subject to cultural bias arising from previous >> exposure.
>>
>> Then I think that "consonance" is too subjective to reduce to a >> single
>> psychoacoustic percept. And I also think that harmonic concordance is
>> too far removed from consonance to use as a basis for theory, as
>> people can get used to mavila and hear all sorts of consonances in >> it.
>>
>> My current obsession is to try and find some kind of pattern in how
>> this type of consonance works. It has something to do, I think, with
>> Rothenberg's idea that people build up mental reference frames or >> maps
>> from scales that they've learned. The subjective component to this
>> would come from what maps people have learned so far and which one
>> they haven't, so if we assume people have "learned" how to place
>> things in a diatonic/5+2 sense, we can make predictions on what >> they'd
>> find "consonant." I also think that Paul had some more of the puzzle
>> figured out with his 22 paper, and maybe some other author I haven't
>> read yet has it all figured out. But either way there is still some
>> type of consonance emerging from "that," and that's what I'd love to
>> have figured out. And what's wrong with that?
>>
>> -Mike
>>
>
> Charles Lucy
> lucy@...
>
> -- Promoting global harmony through LucyTuning --
>
> For more information on LucyTuning go to:
>
> http://www.lucytune.com
>
> LucyTuned Lullabies (from around the world) can found at:
>
> http://www.lullabies.co.uk
>
>
>
>
>
>
>
>

Gene>> But what about things like explaining the consonance (to many people,

>> apparently NOT just me) of 15/8 vs. the lower limit nearby 9/5 and 11/6 or
>"Not to me. I think it would help if we had some actual test data, because you
>keep making assertions about what you think as if they will probably be what
>other people think, and yet often what you claim doesn't make sense to me."

It's based on information I've gathered from other people...where do you get
off saying it's just from me?! When I say "NOT just me" shouldn't it be pretty
obvious I HAVE surveyed people other than myself to get that result? :-S
Specifically I recall MikeB and Kraig Grady...not to mention a handful of
musicians I know personally, favored the 15/8 over 11/6 and 9/5.
Believe me, it's not personal bias as in "wanting it to be that way"...in
fact I wish the 11/6 were the best as it would make 7TET-like scales more
consonant...but the overwhelming evidence I've gathered so far says it's not.

>"You say that 11/6 sounds more consonant than 15/8? In what sense do
you mean "consonant?"

This must be the 30th time I'm saying this. By consonance I mean
SOUNDING/FEELING relaxed regardless of psychoacoustic phenomena. CONCORDANCE
(WHICH I'M NOT USING) would mean psycho-acoustically well balanced. The point
is you'd THINK 11/6 would feel more relaxed because it has better psychoacoustic
properties (IE 11/6 is more CONCORDANT, so far as periodicity, for example), but
it doesn't making 15/8 more CONSONANT...and the same seems to hold for 9/5. I
can swear Carl has spelled out this difference between consonance and
concordance countless times...and EVEN while following his definitions people
seem to find reasons to say I'm the one assuming the wrong definitions. :-S

I must stress again, if a psychoacoustic theory says a dyad should be better
balanced and my ears says otherwise and then I blind-test at least a few
musicians and listeners on it and they unanimously point the the theory being
wrong in that case, I won't be scared to say "the theory has a flaw in this
case". Ears trump what any one psychoacoustic theory says something "should
sound like"...can't we at least agree on that?

Gene>"No, I object to your constant claims about which dyads are more consonant.
based on your own often idiosyncratic reactions."

We if I wanted something that would prove easy to work with and things were
"how >>I<< wanted them to be"...I would
A) Assume the entire range from 9/5 to 15/8 is valid for a good seventh (think
of how many more scales I could make without hitting a sour dyad if that
worked!)
B) Use harmonic entropy as an excuse and assume things like 40/27 would work as
more consonant than something like 5/4 simply because of "the tonal gravity
around 3/2". It's a lot easier to get a 40/27 out of a scale than, say, a 22/15
or a 14/9....
C) Assume odd-limit always works, and be able to correctly assume that chords
such as 16/11 will always sound more relaxed than 22/15, for example (note 22/15
was NOT my idea, Igs introduced me to it back when I thought 16/11 'should be
better', I tried 22/15 on my ears, and they agreed 22/15 was better). Lord
knows that would save me a lot of mathematical work...and perhaps enable me to
just go around merrily calculating things in Tenney Height, for example, without
worrying about more calculations.

Believe me...there is virtually NOTHING about my findings that make my life
more convenient when making scales or more like what I want them to be. In fact
they make things much harder. Plus, as I've said time and again, I constantly
test what I come up with as answers against listeners meaning...people other
than myself. Sometimes my girlfriend or friends will walk into the room,
without seeing what of my compositions I'm playing on my PC and randomly say
"that sounds good" or "that sounds terrible" when I'm just testing a dyad. And
that's NOT my "doctored" opinion, that's hers (un-asked for, even!).

So, unless you think that my ears consistently give odd results and any
people who happen to hear must also have odd ears...I don't see how my work can
be pigeon-holed as Idiosyncratic.

>"But I get tired of hearing what are apparently personal preferences elevated to
>a status I don't see they deserve"
Well...they aren't "just personal" if other people can confirm them.

MikeB>"I explained exactly how it's analogous right after that sentence.
4:5:6:7 is less complex and fuses better, and 8:10:12:15 has some
consonant quality that emerges from the diatonic map that I have. And
the latter quality emerges even if the thirds aren't intoned
differently. Likewise, 11:6 is less complex and fuses better than
15:8, but the latter draws from the same diatonically consonant
quality that makes 8:10:12:15 so consonant."

It seems as if, simply put, to different people each dyad will be the most
consonant. And, groups of people thinking 15/8 is the most consonant and other
groups thinking 9/5 is...is not an impossibility and it does not mean people in
a certain group must all have "personal idiosyncrasies".

Let me pose this another way...can we at least say (yes, including those
people "used to the diatonic map")...that there no one of these three dyads that
has a monopoly over consonance and it's a fair question why 9/5 doesn't
instantly "win all battles"?

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> This posting is copyright Charles Lucy 2010, and explains how this
> consonance/dissonance conundrum actually works for Western harmony.

Wow, does your humility have no limit?

> If you want to understand Western harmony and jazz you need to use a meantone-type
> model

What if I want to understand Mavila harmony, or Porcupine, or Pajara? Or strict JI? Or Indonesian gamelan? I'm writing a friggin' book about EVERY EDO from 5-EDO to 37-EDO! Only one of those is even remotely close to LucyTuning, and that's 31-EDO. You could not possibly have formulated something more useless to me if you had tried.

You accuse people here of being "JI-obsessed", but you sir are vastly more insular. You cannot even see past one form of temperament--your "LucyTuning", which is a natural mathematical phenomenon discovered first by John Harrison that you've had the audacity to name after yourself and claim as your own property (since you did "all the work" to re-discover work done by SOMEONE ELSE). Your words are always suspect, Mr. Lucy. Especially because your crown jewel of a tuning does not even get results that are as well-in-tune as the optimized Meantone temperaments discovered by folks here--folks who *gasp* are humble enough to recognize their discoveries are of natural phenomenon that no man has the right to treat as "property".

Get off your high horse.

> If you want to understand Western harmony and jazz you need to use a
>meantone-type
>
> model

Who said meantone generates the phenomena in Western music in the first
place? JI diatonic, which is not created the same way, can also do the
same...as can many MOS scales with the same general LLsLLLs pattern.

>"You accuse people here of being "JI-obsessed", but you sir are vastly more
>insular. "
I agree JI does not cover nearly everything...and sure almost everything can
be summarized in very high limit JI (think over 11 limit)...but at that point,
is it really even JI? Now saying everything be summarized in mean-tone covers
much less variation than Extended JI and perhaps also less than even 5-limit
JI. Since Lucytuning is mean-tone...it seems obvious it would be more
restrictive "even" than JI. And both JI and Lucytuning throw by the wayside
much of the experimentation with recurrent sequences and, as Igs mentioned,
different TET scales and the whole concept of temperament.

Lucytuning is good for what it is...a solid mean-tone system...but nothing
more. And copyrighting something originally created by someone else and then
saying its view of consonance solves everything...I strongly is far more causing
more confusion in the music world than innovation.

>"Especially because your crown jewel of a tuning does not even get results that
>are as well-in-tune as the optimized Meantone temperaments discovered by folks
>here"
Look at the least-squares diatonic scale and 7-tone scale I made based around
irregularly tempered mean-tone. I'm pretty sure both those options are more
pure, on average, than Lucytune...at least for JI dyads. But if you think
LucyTuning can beat them in terms of consonance...I'd be quite interested to
hear what your LucyTuned "minimax" alternative to them is.

No joke Chris. I have had so many of my ideas, writings, inventions etc. poached by others that this notice is to register my interest in my work.

On 8 Sep 2010, at 13:36, Chris Vaisvil wrote:

> On Wed, Sep 8, 2010 at 6:18 AM, Charles Lucy <lucy@...> wrote:
>
> This posting is copyright Charles Lucy 2010,
>
> --------------------------
>
> This is a joke, right?
>
> Chris
>

Charles Lucy
lucy@...

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can found at:

http://www.lullabies.co.uk

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> It's based on information I've gathered from other people...where do you get
> off saying it's just from me?!

I think it's based purely on cultural conditioning.

> Specifically I recall MikeB and Kraig Grady...not to mention a handful of
> musicians I know personally, favored the 15/8 over 11/6 and 9/5.

Not me, I think 9/5 is the clear winner.

> Believe me, it's not personal bias as in "wanting it to be that way"...in
> fact I wish the 11/6 were the best as it would make 7TET-like scales more
> consonant...but the overwhelming evidence I've gathered so far says it's not.

Evidence??? You've presented a few people you claim agree with you.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
Perhaps we can simply say a lot
> of this is just a side-effect of this "diatonic map" that keeps being brought
> up...and perhaps only applies the dyads consonance as perceived in diatonic
> scales (and is valid in those kinds of scales only!)
>

If you just LISTEN, 15/8 is a bit harsher than 9/5. Preferring it I am pretty sure is some kind of cultural conditioning, like the preference for 12et tunings over more nearly just ones for the 5 and 7 limits which is extremely common and clearly not based on psychoacoustic factors absent conditioning.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> So, unless you think that my ears consistently give odd results and any
> people who happen to hear must also have odd ears...I don't see how my work can
> be pigeon-holed as Idiosyncratic.

Because it's not "work" in the sense of having any scientific validity. It's no more valid than my own personal listening tests. "Work" would be a double-blind listening test, ideally cross-cultural. If a group of Egyptian subjects gives the same answers as another group from Des Moines, that would be worth noting.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"But I get tired of hearing what are apparently personal preferences elevated to
> >a status I don't see they deserve"
> Well...they aren't "just personal" if other people can confirm them.

But your random anecdotes don't count. If you seriously intended to present this as data rather than personal observation and opinion to be taken with a big grain of salt, you'd need a lot more.

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

> If you just LISTEN, 15/8 is a bit harsher than 9/5.

Absolutely. -Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> You accuse people here of being "JI-obsessed", but you sir are vastly more insular. You cannot even see past one form of temperament--your "LucyTuning", which is a natural mathematical phenomenon discovered first by John Harrison that you've had the audacity to name after yourself and claim as your own property (since you did "all the work" to re-discover work done by SOMEONE ELSE).

I wouldn't call it a "mathematical phenomenon". If you pick any irrational number such as pi, you can form a field Q(pi) which extends the rational numbers Q but has the same property of being dense in the reals. Hence there will always be numbers such as 1/2+1/(4pi) octaves which can serve for a meantone tuning. I can propose that everyone should use SmithiTuning, which uses 17/92 pi octaves, or 696.614 cents, as a generator. I could then become upset when people point out it is close to 31et, and even closer to 1/4 comma meantone, and proclaim SmithiTuning is the answer to the mystery of consonance.

On Wed, Sep 8, 2010 at 2:51 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> I wouldn't call it a "mathematical phenomenon". If you pick any irrational number such as pi, you can form a field Q(pi) which extends the rational numbers Q but has the same property of being dense in the reals.

How would a rational number be generated from something like pi, being
as it's a transcendental number...? Also, when you say dense in the
reals, do you mean that its cardinality is aleph-1?

> Hence there will always be numbers such as 1/2+1/(4pi) octaves which can serve for a meantone tuning. I can propose that everyone should use SmithiTuning, which uses 17/92 pi octaves, or 696.614 cents, as a generator. I could then become upset when people point out it is close to 31et, and even closer to 1/4 comma meantone, and proclaim SmithiTuning is the answer to the mystery of consonance.

I thought it was Marcel-JI that was the answer to the mystery of consonance.

-Mike

Hi Charles,

I can appreciate wanting to have credit for your work. I had a truly
patentable idea stolen from me in my working life and it sucked.

However, I doubt that your copyright notice will deter anyone low enough to
steal your ideas and present it as their own.

Chris

On Wed, Sep 8, 2010 at 8:47 AM, Charles Lucy <lucy@...> wrote:

>
>
> No joke Chris. I have had so many of my ideas, writings, inventions etc.
> poached by others that this notice is to register my interest in my work.
>
> On 8 Sep 2010, at 13:36, Chris Vaisvil wrote:
>
>
>
> On Wed, Sep 8, 2010 at 6:18 AM, Charles Lucy <lucy@...<lucy%40harmonics.com>>
> wrote:
>
> This posting is copyright Charles Lucy 2010,
>
> --------------------------
>
> This is a joke, right?
>
> Chris
>
>
> Charles Lucy
> lucy@...
>
> -- Promoting global harmony through LucyTuning --
>
> For more information on LucyTuning go to:
>
> http://www.lucytune.com
>
> LucyTuned Lullabies (from around the world) can found at:
>
> http://www.lullabies.co.uk
>
>
>
>
>
>
>

Gene>"Evidence??? You've presented a few people you claim agree with you."

Well, where's the "easily accessible groups of hundreds of people ready to
take my survey" you seem to indirectly imply I should magically call on?!

There lies the double-standard. If you EXPECT people to be able to come up with
huge surveys to back up their arguments OR ditch them as "purely personal and
not of any potential value to the greater community" be prepared to hand such
people an idea of how to initiate such surveys AND/OR point to references of
someone WITH such access which happens to prove the exact same point WHILE
saying your point is original (ALA LucyTuning referencing Harrison...which ends
up, of course, looking like anything but original).

Which of course...is not reasonable for you to be able to manage to do
either. So I'm a person without a huge base of people from which to run a
survey to "prove it" to the degree you want. Most of us are in that same
boat...and yet it seems you seem very eager to "blame the non-professor for not
having students". Well duh. And lots of people on that list make that same
mistake and blame others for not being able to "make the cake without the
dough".

Feel free to prove me wrong...but you seem to be handing out complaints
which can only be "resolved" through impossible to obtain resources. How on
earth do you expect that assault to be turned into something productive?

>"If you just LISTEN, 15/8 is a bit harsher than 9/5. Preferring it I am pretty
>sure is some kind of cultural conditioning, like the preference for 12et
>tunings over more nearly just ones for the 5 and 7 limits which is extremely
>common and clearly not based on psychoacoustic factors absent conditioning."

Cultural conditioning or not...I'm not going to argue with my ears favoring
the 15/8. I don't see how anyone in the public would say "oh it sounds bad, but
I'm going to keep telling myself it sounds good just because it fits into a
well-established formula". You can't force me to hear something the way you
want it! IMVHO, you prefer 9/5 and I prefer 15/8....and it is likely that
different people do agree with both your and my opinion on that. It's not an
all or nothing deal, both views can have a following, hence why music is an art.

>"Because it's not "work" in the sense of having any scientific validity. It's
>no more valid than my own personal listening tests. "
So where on earth is obvious access to all these people you want me to survey to
prove my results valid? What's your obvious alternative? You don't seem to
have any intention of giving one...but every intention on whining about it. Can
I make a simple point about "I think, and was wondering if anyone else agrees?"
without getting some absurd attack about why I didn't run a huge survey first?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
Can
> I make a simple point about "I think, and was wondering if anyone else agrees?"
> without getting some absurd attack about why I didn't run a huge survey first?

Yes, and I suggest you start, and stop presenting your personal reactions as laws of nature.

>> Well...they aren't "just personal" if other people can confirm them.
>But your random anecdotes don't count.

And if someone else got a few people to agree with your ideas they would
(because they come from YOU (not-random) not me (random) )?
Now it seems you're just discounting my opinions because you don't like them,
regardless of what I do. What are my obvious options? You give me none accept
"give up; since your ideas don't have past papers backing them up or huge
surveys as evidence no one will ever take them seriously enough to, say, help
you get access to what you need to prove anything as scientific".

>"If you seriously intended to present this as data rather than personal
>observation and opinion to be taken with a big grain of salt, you'd need a lot
>more."
A lot more what? A lot more people's academic papers to back up any claims?
What if it's on a subject where I can't find any papers?
For example, the consonance of 22/15 vs. 16/11...of which Igs handed me 22/15
as more relaxed sounding. There's no paper on this...yet if I say Igs and I
prefer 22/15 it has "no basis". Ok, no basis in past research, but so what?
Plomp and Llevelt didn't write their research based on a mathematical formula
based on past research either....but, ah yes, they had a bunch of people to
survey...which makes their "random personal opinion" OK as scientific. Seems
the only option (correct me if you have another) is just a large survey, which I
don't have resources to run...and yet I'm getting "spanked" for "lack of effort
to run the survey?!" Boo.....

>"I thought it was Marcel-JI that was the answer to the mystery of consonance."
There is no one answer....yet there is a sad propensity to complain about anyone
on the list saying they even have part of an answer unless they also claim to
solve anything just as much as, and often more than, those who claim they have a
whole answer.
Discussions on Harmonic Entropy and Critical Band Dissonance seem accepted
as the formula covers all intervals (and thus in a way claims to summarize
everything).
Meanwhile anything that comments on say, preference of one dyad over another
that does NOT match the results in huge formulas like those (whether by myself
or others), seems to get shot down very quickly and gets little to no options by
which to develop or be tested on a larger population.

No wonder people seem to "only" claim they have everything or
nothing....we've created a "game" where people are encouraged to say they have
everything on order to be taken seriously....even when they are not even close!

Gene>"Yes, and I suggest you start, and stop presenting your personal reactions
as laws of nature."

Now you may not agree with what my ears hear, but how on earth have I presented
such things as laws of nature?!
Or do you think any statement in music must somehow be explained by some "laws
of nature" to be anything more than utterly random?
Suppose I'm an artist and I make a drawing. Do I need a scientific explanation
of the shift of colors to prove it's worth anyone's paying attention to?

I have not once said "this is how it must work" or anything along those
lines. My statements are always in a form such as "I was wondering why the 15/8
seems more CONSONANT than the 11/6?".
I never say anything like "15/8 is always more CONCORDANT than 11/6"...or
something that would imply a numerical law of nature.

LOOK...if I SAY "CONSONANT", I MEAN RELAXED SOUNDING to the listener
(regardless of how IE some people use different psychoacoustic phenomena in
different weights, some use cultural conditioning, some use other things). At
most, I mean "consonant to a good number (NOT MEANING ALL!) listeners". To be
a law of nature would imply "all". All objects are effected by gravity, etc.

If I said CONCORDANT, I would mean "this is relevant to a scientific law of
psycho-acoustics"...the only time I use that term is comparing things like
intervals under 7/6 and describing the beating of the root tone becoming more
and more intense as they become closer. Which, of course, can be graphed and
fits into wave theory. So, of course, does periodicity. I can take two sine
waves, mix them together, and show that lower numbered fractions take less long
to repeat than higher-numbered ones on an actual graph of the sound waves. Its
physics.

Now when I make statements on findings my ears point out to me, I'm talking
CONSONANCE NOT CONCORDANCE. Although it seems no matter how many times I say
this I get whined at for claiming my observations are "about concordance". Why
the problem? Can't you understand I'm not stating anything as a law of
nature...but simply saying somethings don't sound like what psychoacoustics say
they should sound like?

If you can't get it through your head at all that when I say consonance I do
NOT mean concordance, that's your issue of lack of understanding, not mine.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> I have not once said "this is how it must work" or anything along those
> lines. My statements are always in a form such as "I was wondering why the 15/8
> seems more CONSONANT than the 11/6?".

Which is meaningless as an objective statement, as objective consonance has no known formulation. Things which don't need to resolve in a particular style, or things you personally like, is hardly a good basis.

> I never say anything like "15/8 is always more CONCORDANT than 11/6"...or
> something that would imply a numerical law of nature.

The difference is, "concordance" is attempting to locate something actual, or several related things actual.

>
> LOOK...if I SAY "CONSONANT", I MEAN RELAXED SOUNDING to the listener

There is no "the listener". There is only "a listener". So this isn't really a definition. And do you really think 15/8 sounds relaxed?

Can't you understand I'm not stating anything as a law of
> nature...but simply saying somethings don't sound like what psychoacoustics say
> they should sound like?

What makes you think psychoacoustics has anything to say about something so contextual and personal as "relaxed sounding"?

Chris wrote:

> However, I doubt that your copyright notice will deter anyone low
> enough to steal your ideas and present it as their own.

Carl's 478th law: Those most concerned with IP protection have
the least IP worth protecting.

-Carl

Gene>"Which is meaningless as an objective statement, as objective consonance
has no known formulation. Things which don't need to resolve in a particular
style, or things you personally like, is hardly a good basis."
Basis for what? Do you think we should, for example, need an equation to
express why Beatles songs "worked" in order to make a statement "this music just
might have some useful patterns within it...useful to a good amount of
people"? It seems obvious to me musical students don't study say, Beethoven,
because the math works out so prove-ably. Instead they are captivated by the
music and wonder if they can find any patterns that can be translated into math
and then used elsewhere.

When I brought up the 15/8 my point was it seemed to work well for me and
apparently most of the people I was able to try it on (even vs. 11/6 and 9/5).
I was hoping for answers like "yes, there's a psychoacoustic pattern (that you
didn't know about yet) that explains this" or "no, but it's worth looking into
to perhaps develop". The only real productive answer I believe I got from the
whole thing is "it might just be a response of cultural conditioning...and
favoring of a learned diatonic framework", which isn't exactly the most useable
to help, say, form new scales or dyads with similar and original
properties...but at least it was a shot.

>"The difference is, "concordance" is attempting to locate something actual, or
>several related things actual."
Who reserves the godly right to define actual? It sounds like the old
argument that phenomena in quantum physics aren't real until they are measured
(but who can measure them?). If by actual you mean existing in Physics as
measurable by anyone (IE via a graph of soundwaves) I agree...otherwise that
statement seems scrambled.

Me>>Can't you understand I'm not stating anything as a law of
>> nature...but simply saying somethings don't sound like what psychoacoustics say
>>
>> they should sound like?
>"What makes you think psychoacoustics has anything to say about something so
>contextual and personal as "relaxed sounding"?
Well you seem to be saying "oh, well your statements are pointless because
they lack direct numeric evidence". So what does have direct numeric evidence?
Psychoacoustics, specifically signal processing phenomena. Therefore I am
assuming you are saying "you need your results to be based on
psychoacoustics...otherwise they are just subjective"...that and "relevant ONLY
to you unless you find a huge number of people to support your findings".

Now I'm responding "but they can be relaxed sounding WITHOUT being
scientifically proven as such via psychoacoustics...and saying something is
relaxed sounding without relating psychoacoustics does not automatically make
that theory a random personal guess regardless of personal effort".
Can you say it may be biased? Sure....but you seem to be going on and on
about how it must be 100% biased and the efforts I've made to at least do small
surveys (IE of what resources are available to me)...and that's simply not fair
or useful. It's whining for the sake of whining by any other words.

>"Carl's 478th law: Those most concerned with IP protection have
the least IP worth protecting."

Ah, only 478?! You need more, the Jews had 613....SERVED! :-D

Hi Carl,

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

> Paul did a lot of work with deep and experienced listeners
> like George Secor, Margo Shulter, and many many others as he
> was experimenting with harmonic entropy.
> The Vos curve version is based on the work of Joos Vos, who
> did experiments in which listeners rated the purity of
> mistuned intervals. I don't have the paper that Paul had
> so I can't comment further, but I believe Vos found that
> purity ratings obeyed an exponential distribution
> http://en.wikipedia.org/wiki/Exponential_distribution
> around just ratios. (Please note that this is NOT the same
> as taking the exponential of entropy at the end, as you've
> been doing.)

Gotcha.

> Regular h.e. uses a Gaussian distribution instead, which is
> the most natural assumption and the one used by Goldstein in
> his famous 'central pitch processor' model.

Interesting. I always wondered about how he figured out the distribution.

> Those are the two principle versions. Beyond that, it's
> mainly how much noise you want to assume (the variable "s")
> and whether you want to take the exponential when you're
> done. Goldstein deduced that s was about 1%, but Paul tried
> values between 0.6% and 1.2%. Making it smaller causes more
> local minima to appear.

So decreasing "s" increase the number of local minima, but otherwise does not alter the distribution around the minima, or the depth of the minima?

> > Mike mentioned an octave-equivalent version of H.E. I'm
> > curious how that works,
>
> I think Paul just averaged the entropy of every pair of
> inversions. Like 16/15 and 15/8 would always be the same.
> So generally it will lead to worse results. It's only
> useful if you insist on using octave equivalence in your
> music theory.

Well, it might be more useful in ranking EDOs, because at any rate you are always going to have octave inversions. But how would that formulation deal with, say, 15/4 vs. 15/8? Did he average in higher-octave ratios? (Or is that even necessary?)

> > I have a hard time hearing 12/5 as being half as "restful"
> > as 6/5, or hearing 5/2 as twice as "restful" as 5/4.
>
> In both cases it's not twice, but root twice.

I thought Tenney Height=n*d? 6*5=30, 12*5=60, 60/30=2. Or is Tenney Height sqrt(n*d)?

> I definitely
> hear 5/2 as sweeter than 5/4. I was dubious about 6/5 vs
> 12/5 but you know, I just tried it in Scala in two different
> octaves and I have to say, to my surprise 6/5 sounds more
> concordant.

Okay, I'll admit I agree about 5/2 vs 5/4 (though 5/1 vs 5/2 is harder to say). But I tried 6/5 vs 12/5 and I don't agree. I tried it with a sawtooth and a piano in Logic, FWIW.

I said:

> > It just boggles me that some intervals go up geometrically
> > in Tenney Height as you increase the octave, while others
> > drop geometrically. That just doesn't fit with my experience
> > of concordance.

Carl said:

> To make 5/2 in a harmonic series, you use lower harmonics
> than to make 5/4. And same with 6/5 and 12/5.

Well, then this a weakness in using the harmonic series as a model for consonance, I should think!

> > It also totally blows chunks all over the "critical bandwidth"
> > theory, right?
>
> I believe that making a dyad wider by an octave always
> decreases roughness according to Sethares' model, but don't
> quote me on that.

Seems correct. It does by Plomp and Levelt's model, doesn't it?

> > So I'm wondering if the octave-equivalent
> > formulation of H.E. solves that somehow.
>
> I don't think so. However you may be on the right track
> because your observations seem to be calling for SOME
> discount on factors of 2.

Ha, it's too bad I don't have the academic training to really come up with my own model. I'm a philosopher, not a scientist. I'll have to be content with "pointing in the right direction" if I can.

> By the way, you should know that I almost can't believe I'm
> having this conversation with you. I was driving -- I think
> it was the week before I left for burning man -- playing my
> iPhone on shuffle through my car, and Numerology came up.
> And I was totally floored. What insanely incredible music!
> I mean, like, you're a genius. Way better at music than I
> am at music theory. Then I had the same f'ing experience
> today when listening to Aaron Johnson's Peer Gynt.
> I mean, damn.

Thanks, Carl! I'm flattered, really. I like to think that "Numerology" was just a "proof of concept" and that some day I'll get around to doing something greater. I have much more grandiose ideas in my head, anyway. But right now, I am occupied by the task of writing this Primer, and I find your theory guidance utterly indispensable. The sooner I get this theory stuff figured out, the sooner the writing will be done, and the sooner I can get back to making music. I've been promising Ron Sword that I'd bump out some more xenharmonic metal soon, but I'm kind of more on a psychedelic kick. But if "Numerology" really does it for you, that's extra incentive for me to do more in that direction!

I'm going to respond to more of this post in another reply.

-Igs

> Paul did a lot of work with deep and experienced listeners
> like George Secor, Margo Shulter, and many many others as he
> was experimenting with harmonic entropy.

So what defines a valid survey population that makes a theory scientifically
valid?

I've heard Plomp and Llevelt used about 30 people for their theory on
roughness. "Many many others", in this statement, goes unnamed (or are we just
to assume certain scholars ears bear a whole lot more weight than the average
person's?!)

There's a lot of complaining about people's theoretical efforts "having no
scientific basis". The result seems to be people name-calling left and right,
making it look like many people are scientifically irrelevant. Isn't there a
better way to go about this that gives people a fair chance to work on
establishing their theories to the best they can (even if that end is "less than
perfect")?

On Wed, Sep 8, 2010 at 2:28 AM, Carl Lumma <carl@...> wrote:
>
> > 4:5:6:7 is less complex and fuses better, and 8:10:12:15 has some
> > consonant quality that emerges from the diatonic map that I have.
>
> 8:10:12:15 is concordant even without a map.

I never said anything about concordance. I said that there is some
type of property which might reasonably also be called "consonance"
that emerges from the map, and that the major 7 has it. The fact that
a major 7 chord is very close to a minimal set for the major scale
might have something to do with it.

> But that's also got NOTHING TO DO with 15:8 vs 11:6 or any other
> comparison of dyads. It may REMIND you of it in some vague
> way that nobody else has a chance of understanding, but
> that's about it.

The fact that shell voicings exist as a compositional technique should
probably indicate that quite a few people have a chance of
understanding them.

> > Likewise, 11:6 is less complex and fuses better than
> > 15:8,
>
> Does it? The two are only marginally distinct. Neither
> ratio is particularly tunable by ear.

I can get them both in the ballpark by ear pretty well.

-Mike

On Wed, Sep 8, 2010 at 10:13 AM, Michael <djtrancendance@...> wrote:
>
> Specifically I recall MikeB and Kraig Grady...not to mention a handful of musicians I know personally, favored the 15/8 over 11/6 and 9/5.

I never said that. I never said that I favor either of them. I said
that 15/8 is consonant in a way that 11/6 is not, and that that
consonance comes from the diatonic map, and that in that sense it
doesn't matter whether it's tuned as a 5-limit major 7 or a 7-limit
supermajor 7. If I had some map that gave 11/6 some kind of function
that was comparable to that of the maj7, then perhaps I'd hear it as
just as consonant.

-Mike

Hi Carl, here's "Part 2":

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
I wrote:
> > As to the list of 7-limit triads, my reckoning is that 31-EDO
> > would dominate all the triads that include 5, and 36-EDO would
> > dominate all the ones that don't.

Carl wrote:

> Ah, but that's why multiply the error by EDO size -- getting
> badness!

Okay, let's pursue this further. Ultimately, the whole purpose of my writing on harmonic properties of EDOs is to reveal effective ways to use them. For instance, I don't see 13-EDO as a bad tuning, I see it as a bad tuning if you want 4:5:6 chords. If you want 8:11:13 chords, it looks pretty good! I know 11- and 13-limit intervals rank low on concordance, but having played a Catler 12-tone Ultra Plus, I can vouch that they can be essentially beatless if they're well in-tune, which to me marks them as useful. So let's ignore the issue of octave equivalence for now and assume that there is something to a fit to the harmonic series--it may not be an absolute guide to concordance, but in a given EDO where options are limited, it seems to be the case that the chords that fit lowest in the series sound the strongest.

So: let's suppose we want to look at n-limit triads and how they are approximated in various EDOs up to 37. Now, let's assume that our motivation here is not to rank EDOs, but to demonstrate their strongest harmonic possibilities, so our "badness" measure will be on a triad-by-triad basis. What I believe we need to decide is 1) what counts as an approximation, and 2) what our limit is. These two questions are obviously interrelated. Let's presume we're sticking with odd limit rather than prime limit, for simplicity's sake.

Now, if we set our odd limit at a low number, like 7, we can be sure that any low-error approximation will definitely sound concordant, but we will have very few triadic possibilities; and unless we set a wide margin of error, many EDOs won't be able to be said to approximate any of them, so it won't be a good "guide". If we set our odd limit too high, we might end up with one tempered chord approximating multiple ratios if our error threshold is not tight enough. For instance, at an odd limit of 23, the 12-tET minor triad can be said to approximate both 10:12:15 with a high error, and 16:19:24 with a low error; I predict it will be hard for people to interpret what exactly that means (even I find it rather confusing).

So, uh, what's the best way to go about this? Is it really even do-able?

-Igs

Igs wrote:

> So decreasing "s" increase the number of local minima, but
> otherwise does not alter the distribution around the minima,
> or the depth of the minima?

I believe it makes the minima a little sharper also. You
can browse the plots on harmonic_entropy and see for yourself.

> Well, it might be more useful in ranking EDOs, because at any
> rate you are always going to have octave inversions.

Yes, they will be there, but that doesn't mean they're
equally concordant!

> But how would that formulation deal with, say, 15/4 vs. 15/8?
> Did he average in higher-octave ratios? (Or is that even
> necessary?)

I dunno.

> I thought Tenney Height=n*d? 6*5=30, 12*5=60, 60/30=2.
> Or is Tenney Height sqrt(n*d)?

Strictly speaking, the former. But the latter is what you
should use for this business, and I don't have a name for
it, so I call it Tenney height (because I'm a rebel, remember?).

> Okay, I'll admit I agree about 5/2 vs 5/4 (though 5/1 vs 5/2
> is harder to say).

Ah, but 5/1 is out of the ~ 3 octave range whence Tenney
height works. Remember there are always two caveats to
Tenney height, or any ratio-based rule: TOLERANCE and SPAN
(usually spelled all-caps). TOLERANCE is why we don't go
above a product of ~ 70. SPAN is why we generally can't
rate intervals larger than 3 octaves.

> But I tried 6/5 vs 12/5 and I don't agree. I tried it
> with a sawtooth and a piano in Logic, FWIW.

Noted.

> > I believe that making a dyad wider by an octave always
> > decreases roughness according to Sethares' model, but don't
> > quote me on that.
>
> Seems correct. It does by Plomp and Levelt's model, doesn't it?

For two sines, yes, discordance just trails off to zero
after the maximum. So I don't see how Sethares' model
can do anything else.

> Thanks, Carl! I'm flattered, really. I like to think that
> "Numerology" was just a "proof of concept" and that some day
> I'll get around to doing something greater.

You must have had a drummer. WTF was you lineup back then!?

> But if "Numerology" really does it for you, that's extra
> incentive for me to do more in that direction!

Oh, it's not the only one of yours that does it for me.
Just the one that came up on shuffle the other day.
And when I wrote the above I listened again to make sure
it wasn't an emotional fluke on my part... nope!

Actually I tend to like gentler music, which often makes
it harder for me to objectively listen to metal. But with
a track like Numerology, there's never a problem!

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Igs,
>
> > > 1. Sethares' dissonance curve seems to have the same problem!
> > > Zoom in on fig. 3 here:
> > > http://eceserv0.ece.wisc.edu/~sethares/paperspdf/consonance.pdf
> >
> > Yeah, that graph is hard to read. Really I'd want to find a
> > way to put on a "0 to 100" scale the same way I did with H.E.;
> > that's the only way I can *really* compare them.
>
> Yes, that'd be nice. But it's evident that it has the same
> problem. The tick marks are every 100 cents, so 720 is about
> a fifth of the way across.
>
> > > 2. Vos-curve harmonic entropy should ameliorate the problem
> > > somewhat. Have a look at:
> > > /tuning/files/dyadic/secor4.gif

See below for details on how the curve in that figure was arrived at.

> > Dag nabbit, how many H.E. curves are there??
>
> Paul did a lot of work with deep and experienced listeners
> like George Secor, Margo Shulter, and many many others as he
> was experimenting with harmonic entropy.

The curve in file secor4.gif is the result of a discussion between Paul & myself on the harmonic entropy list, beginning with message #600 (1 Jul 2002). This was a continuation from msg. #38387 on the main list. If you follow the discussion, you'll see that the curve was tweaked to conform as nearly as possible with some listening tests I made using a retuned electronic organ in the early 1960's to determine points of maximum consonance (which correspond to simple-number ratios) and also of maximum dissonance (which was admittedly rather subjective, but better than blind guessing) between neighboring consonant intervals.

Paul also supplied me with the raw data for each cent from 0 to 1200 cents in the above curve; which will allow you to compare the entropy value for one interval (rounded to the nearest cent) to another. Go to this folder in the files section:
/tuning/files/secor/
and find the spreadsheet "raw-entr.xls".

I was also interested in converting the raw entropy values (col. C) to "sonance" values (col. B), where 1:1 would be exactly 1, and other intervals would be something between zero and 1, with the most consonant intervals having higher values (2:1 has 0.63212), and the global maximum (at ~67 cents) having the lowest value (0.02851). Col. D shows local maxima & minima, and ratios are identified in col. E.

After making this table, I added some cells to calculate metastable intervals, which frequently occur close to local HE maxima.

I haven't been following your discussion, since I have had very little free time lately. I drop in here every couple of days to check whether there's anything I might reply to by searching for my name and for "sagittal", which is how I found this message. I hope that this has been of some help.

--George

Hi Carl,

I said:
> > Well, it might be more useful in ranking EDOs, because at any
> > rate you are always going to have octave inversions.

You said:
> Yes, they will be there, but that doesn't mean they're
> equally concordant!

No, but then again if you play in these EDOs on a piano or guitar and have any octave-doubling, you're going to be stuck with the inversions anyway. I figure that in practice, the difference in concordance between octave inversions probably averages out anyway. But it's still moot for the time being, because I've persuaded myself that this isn't the right model for my purposes.

> Ah, but 5/1 is out of the ~ 3 octave range whence Tenney
> height works. Remember there are always two caveats to
> Tenney height, or any ratio-based rule: TOLERANCE and SPAN
> (usually spelled all-caps). TOLERANCE is why we don't go
> above a product of ~ 70. SPAN is why we generally can't
> rate intervals larger than 3 octaves.

Ah, I think I must have missed that part when the concept was introduced to me.

> > Thanks, Carl! I'm flattered, really. I like to think that
> > "Numerology" was just a "proof of concept" and that some day
> > I'll get around to doing something greater.
>
> You must have had a drummer. WTF was you lineup back then!?

No drummer, just my trusty old "Ultimate Sound Bank" PlugSound Drums VST and Cubase SL3. Crikey, if I could find a drummer who not only could play a 5:6:7 triple polyrhythm, but accent every time each pair of meters shared a down-beat, there'd be no stopping me! I once knew a drummer who might have been capable of that, but I haven't spoken to him in years (and last I did, he didn't think my guitar chops were quite up to snuff). No, all my microtonal music up to this point is strictly solo work--me and a computer. "Map of an Internal Landscape" was all point-and-click, too; I only recently invested in a MIDI keyboard.

> Oh, it's not the only one of yours that does it for me.
> Just the one that came up on shuffle the other day.
> And when I wrote the above I listened again to make sure
> it wasn't an emotional fluke on my part... nope!

Well, I probably put more compositional work into that song than 95% of the rest of my output. At the very least, your reaction should be incentive for me to be more compositionally-intensive. Though I've got a bunch of heavily-improvised micro stuff coming down the pipe, I'll be curious as to how you like it. I certainly don't take your reactions lightly, since you are so gosh-darn knowledgeable!

-Igs

Michael wrote:

> So what defines a valid survey population that makes a theory
> scientifically valid?

Not on-topic.

> "Many many others", in this statement, goes unnamed (or are we
> just to assume certain scholars ears bear a whole lot more
> weight than the average person's?!)

Troll.

-Carl

Mike wrote:

> > But that's also got NOTHING TO DO with 15:8 vs 11:6 or any other
> > comparison of dyads. It may REMIND you of it in some vague
> > way that nobody else has a chance of understanding, but
> > that's about it.
>
> The fact that shell voicings exist as a compositional technique
> should probably indicate that quite a few people have a chance
> of understanding them.

Did you put the wrong text here? It doesn't seem to relate
to the >> text.

> > Does it? The two are only marginally distinct. Neither
> > ratio is particularly tunable by ear.
>
> I can get them both in the ballpark by ear pretty well.

How do you know?

-Carl

Igs wrote:

> I figure that in practice, the difference in concordance
> between octave inversions probably averages out anyway.

We've seen that such an approach fails for extended JI.
It may be possible to mine material that works this way
out of the first 37 EDOs though, so it may still be
useful to you.

> > You must have had a drummer. WTF was you lineup back then!?
>
> No drummer, just my trusty old "Ultimate Sound Bank" PlugSound
> Drums VST and Cubase SL3. Crikey, if I could find a drummer
> who not only could play a 5:6:7 triple polyrhythm, but accent
> every time each pair of meters shared a down-beat, there'd be
> no stopping me!

Indeed! Did you know the Astroid Power-Up drums were basically
played on the spot by Deantoni Parks?

-Carl

Igs:

> Okay, let's pursue this further. Ultimately, the whole purpose
> of my writing on harmonic properties of EDOs is to reveal
> effective ways to use them. For instance, I don't see 13-EDO
> as a bad tuning, I see it as a bad tuning if you want 4:5:6
> chords. If you want 8:11:13 chords, it looks pretty good!
> I know 11- and 13-limit intervals rank low on concordance, but
> having played a Catler 12-tone Ultra Plus, I can vouch that
> they can be essentially beatless if they're well in-tune, which
> to me marks them as useful. So let's ignore the issue of
> octave equivalence for now and assume that there is something
> to a fit to the harmonic series--it may not be an absolute guide
> to concordance, but in a given EDO where options are limited,
> it seems to be the case that the chords that fit lowest in the
> series sound the strongest.

No arguments here.

> So: let's suppose we want to look at n-limit triads and how
> they are approximated in various EDOs up to 37. Now, let's
> assume that our motivation here is not to rank EDOs, but to
> demonstrate their strongest harmonic possibilities, so our
> "badness" measure will be on a triad-by-triad basis.

This won't be badness then. Badness is error * complexity
for a temperament. Complexity means "number of notes".
For an EDO, the EDO number is a good kind of complexity.

You are instead suggesting a kind of weighted error. You
want to reward approximate chords both for their low error
and for the concordance they'd have with zero error.

First, you create a list of candidate triads, say, of
product limit < 1000. Then you tune them in the EDO. Then
you rank these by

1 / (cubert(a*b*c) * RMSerror)

or maybe by

1 / (sum of the dyadic h.e.)

> What I believe we need to decide is 1) what counts as an
> approximation, and 2) what our limit is.

By ranking all the possibilities by weighted error, we can
avoid having to choose cutoffs, or at least, reduce the
sensitivity of our results to such choices.

> So, uh, what's the best way to go about this? Is it really
> even do-able?

Of course it's doable. If you like the above I can easily
run some numbers for you.

-Carl

On Wed, Sep 8, 2010 at 6:42 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > > But that's also got NOTHING TO DO with 15:8 vs 11:6 or any other
> > > comparison of dyads. It may REMIND you of it in some vague
> > > way that nobody else has a chance of understanding, but
> > > that's about it.
> >
> > The fact that shell voicings exist as a compositional technique
> > should probably indicate that quite a few people have a chance
> > of understanding them.
>
> Did you put the wrong text here? It doesn't seem to relate
> to the >> text.

The point is that 15:8 doesn't remind me of it in a vague personal
way. It wasn't like when I was a kid, they played 15:8 and maj7 chords
together and hit me with a 2x4 if until I associated the two. It's
that there is a cognitive reason why they are connected, and it isn't
specific to me, but anyone who has an internalized diatonic template,
and have learned through exposure how maj7 chords can be used within
that framework as consonances.

And the cognitive reason is that, for whatever reason, if you
eliminate the fifth in most diatonic tetrads, the sound of the chord
stays mostly "the same." Maybe this has to do with Rothenberg's idea
of minimal sets, maybe because the fifth is an interval giving
relatively less diatonic information, I dunno. But it works and has
been used for a long time.

So if you have a maj7 chord, the bare minimum that you need to make it
sound like a maj7 chord is C-E-B. This could also be an aug^7 chord,
but you're probably not going to assume the presence of an aug5 unless
it's being hinted at.

A shell voicing is like an ultra-bare minimum voicing of 2 notes that
still roughly implies that chord. And when you play C-B on a piano, it
still sort of sounds like a major 7 chord. And it still retains some
of the same musical consonance that a maj7 chord does, for anyone who
has that map. This type of consonance will also pop up if the dyad
fits anywhere within the maj7 range (e.g. 27/14 will still work) and
probably has something to do with Rothenberg equivalence classes,
although I haven't worked it out.

> > > Does it? The two are only marginally distinct. Neither
> > > ratio is particularly tunable by ear.
> >
> > I can get them both in the ballpark by ear pretty well.
>
> How do you know?
>
> -Carl

On my 31-tet nylon guitar, I was having trouble tuning major thirds
precisely just because the timbre wasn't bright enough to give me
enough beating unless I was really far out. In that sense, I can get
15:8 by ear because I can easily get a 12-tet C-B by ear, which will
itself get me "in the ballpark." I can get 11:6 by ear because I can
get 24-tet C-Bv by ear. Or, I can imagine the C as a fifth above F,
and then try to get 11:4 above that F by going to Bv.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> It's that there is a cognitive reason why they are connected,
> and it isn't specific to me, but anyone who has an internalized
> diatonic template

Yes, sorry if you were originally talking about musical
consonance.

> On my 31-tet nylon guitar, I was having trouble tuning major
> thirds precisely just because the timbre wasn't bright enough
> to give me enough beating unless I was really far out. In that
> sense, I can get 15:8 by ear because I can easily get a 12-tet
> C-B by ear, which will itself get me "in the ballpark." I can
> get 11:6 by ear because I can get 24-tet C-Bv by ear.

That's got nothing to do with tuning by ear! If you
can't get a 5:4 there's no way in hell you can get 15:8
or 11:6.

-Carl

On Wed, Sep 8, 2010 at 7:18 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > It's that there is a cognitive reason why they are connected,
> > and it isn't specific to me, but anyone who has an internalized
> > diatonic template
>
> Yes, sorry if you were originally talking about musical
> consonance.

Yep. My main interest now is in delving further into it and figuring
out how it works, since I think it's the most important factor in
determing the end gestalt of "consonance" anyway. More so than
periodic concordance or avoiding roughness. If this weren't the case,
then mavila would be pretty unusable :)

Hence my recent interest in Rothenberg (and Paul's tonality papers),
since I think they have laid the groundwork for some kind of unified
theory of everything with what they've figured out.

> > On my 31-tet nylon guitar, I was having trouble tuning major
> > thirds precisely just because the timbre wasn't bright enough
> > to give me enough beating unless I was really far out. In that
> > sense, I can get 15:8 by ear because I can easily get a 12-tet
> > C-B by ear, which will itself get me "in the ballpark." I can
> > get 11:6 by ear because I can get 24-tet C-Bv by ear.
>
> That's got nothing to do with tuning by ear! If you
> can't get a 5:4 there's no way in hell you can get 15:8
> or 11:6.

That's tuning by ear, isn't it? The best other way of tuning something
by ear that I know is to screw around with the notes until the
partials collide. This approach, for me, fails miserably with timbres
that aren't very bright.

How many cents do you have to be within range to count as having
gotten it? I'll test myself.

-Mike

Mike wrote:

> > > On my 31-tet nylon guitar, I was having trouble tuning major
> > > thirds precisely just because the timbre wasn't bright enough
> > > to give me enough beating unless I was really far out. In that
> > > sense, I can get 15:8 by ear because I can easily get a 12-tet
> > > C-B by ear, which will itself get me "in the ballpark." I can
> > > get 11:6 by ear because I can get 24-tet C-Bv by ear.
> >
> > That's got nothing to do with tuning by ear! If you
> > can't get a 5:4 there's no way in hell you can get 15:8
> > or 11:6.
>
> That's tuning by ear, isn't it?

Absolute pitch, pitch memory, vocal tension pitch, melodic
relative pitch... none of these count.

> How many cents do you have to be within range to count as having
> gotten it? I'll test myself.

You'll need a suitable instrument. Guitar doesn't really
cut it, and piano is too difficult to mess with for
experiments. Harpsichord is ideal. Or a pair of tunable
oscillators.

-Carl

On Wed, Sep 8, 2010 at 7:45 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > That's tuning by ear, isn't it?
>
> Absolute pitch, pitch memory, vocal tension pitch, melodic
> relative pitch... none of these count.

Haha, I dunno how I'm going to get away from them then :) What counts?
Just hearing when the partials collide?

> > How many cents do you have to be within range to count as having
> > gotten it? I'll test myself.
>
> You'll need a suitable instrument. Guitar doesn't really
> cut it, and piano is too difficult to mess with for
> experiments. Harpsichord is ideal. Or a pair of tunable
> oscillators.
>
> -Carl

Alright, I'll mess with a pair of tunable oscillators. What should I
use, a sawtooth wave? Triangle?

-Mike

Mike wrote:

> > Absolute pitch, pitch memory, vocal tension pitch, melodic
> > relative pitch... none of these count.
>
> Haha, I dunno how I'm going to get away from them then :) What
> counts? Just hearing when the partials collide?

harmonic relative pitch

> Alright, I'll mess with a pair of tunable oscillators. What should I
> use, a sawtooth wave? Triangle?

triangle

Try to tune a bunch of intervals. Attempt the list three
times. Blind yourself but record your accuracy each time.
Report results. Profit.

-Carl

On 9 September 2010 06:10, cityoftheasleep <igliashon@...> wrote:

> So, uh, what's the best way to go about this?  Is it really even do-able?

If you want temperaments that approximate 8:11:13, you can go to

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

and put 8:11:13 into the "limit" box. If you want something more
complicated than that you'll probably have to write your own code.

Graham

On Wed, Sep 8, 2010 at 8:07 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > > Absolute pitch, pitch memory, vocal tension pitch, melodic
> > > relative pitch... none of these count.
> >
> > Haha, I dunno how I'm going to get away from them then :) What
> > counts? Just hearing when the partials collide?
>
> harmonic relative pitch
>
> > Alright, I'll mess with a pair of tunable oscillators. What should I
> > use, a sawtooth wave? Triangle?
>
> triangle
>
> Try to tune a bunch of intervals. Attempt the list three
> times. Blind yourself but record your accuracy each time.
> Report results. Profit.
>
> -Carl

OK, how about these intervals:

5/4
11/8
3/2
7/4
11/6
15/8
2/1

Is that good for starters? That's going to be quite a bit of intervals if not :)

-Mike

Hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
Me:
> > So: let's suppose we want to look at n-limit triads and how
> > they are approximated in various EDOs up to 37. Now, let's
> > assume that our motivation here is not to rank EDOs, but to
> > demonstrate their strongest harmonic possibilities, so our
> > "badness" measure will be on a triad-by-triad basis.

Carl:
> This won't be badness then. Badness is error * complexity
> for a temperament. Complexity means "number of notes".
> For an EDO, the EDO number is a good kind of complexity.

Okay, I take back my stipulations, then. You originally suggested a list of the EDOs with the lowest badness for all the 7-limit triads. That got me thinking that for some n-limit, it could be the case that each EDO has at least one triad for which it gives the lowest badness. What I want to do is figure out what that limit is, and then generate a list of all the triads in that limit and which EDO gives the lowest badness for them. If this is not a good idea (i.e. if that limit will be too high, like above the 31-odd-limit), I'd like to know also.

-Igs

Mike wrote:

> OK, how about these intervals:
>
> 5/4
> 11/8
> 3/2
> 7/4
> 11/6
> 15/8
> 2/1
>
> Is that good for starters? That's going to be quite a bit of
> intervals if not :)

Perfecto! -C.

Igs:

> Carl:
> > This won't be badness then. Badness is error * complexity
> > for a temperament. Complexity means "number of notes".
> > For an EDO, the EDO number is a good kind of complexity.
>
> Okay, I take back my stipulations, then.

Don't do that! I'm just trying to figure out what you want.
I thought you wanted to *compare* EDOs, in which case badness
is a good bet. If you want to find resources in each EDO,
weighted error is a good bet.

> You originally suggested a list of the EDOs with the lowest
> badness for all the 7-limit triads.

Yes, but you didn't seem to like it. What do you make of
the suggestion I just sent?

> What I want to do is figure out what that limit is, and
> then generate a list of all the triads in that limit and
> which EDO gives the lowest badness for them. If this is
> not a good idea (i.e. if that limit will be too high,
> like above the 31-odd-limit), I'd like to know also.

I'm not sure I understand. Here's what I know how to do:

Find the best chord for each ET
Find the best ET for each chord

Which path do you choose?

PS- There is an orc behind the oak door.

-Carl

Me>"Specifically I recall MikeB and Kraig Grady...not to mention a handful of
musicians I know personally, favored the 15/8 over 11/6 and 9/5."
MikeB>"I never said that. I never said that I favor either of them. I said that
15/8 is consonant in a way that 11/6 is not,"
Ah ok...now that makes more sense. I took it "in a way" to mean "to a
level"...instead of "in a different way than".

>"If I had some map that gave 11/6 some kind of function
that was comparable to that of the maj7, then perhaps I'd hear it as
just as consonant."
I would be interested to hear such a map... Not saying it's not possible
just, so far, I've tried several different maps full of low-limit intervals and,
in all of those, 15/8 stands out as sounding most consonant to me so far.

Igs>"For instance, at an odd limit of 23, the 12-tET minor triad can be said to
approximate both 10:12:15 with a high error, and 16:19:24 with a low error; I
predict it will be hard for people to interpret what exactly that means (even I
find it rather confusing)."
Interesting, yeah it's a tough nut. If you believe Harmonic Entropy reigns
king, I'd highly suspect you judge it as a 10:12:15 and say the brain "rounds it
to that". But if you think that any couple of dyads within the chord is over 7
cents off pure is usually bad news, go for 16:19:24. Personally I'd say be
careful and state it as 16:19:24, but I'm a "pessimist". :-D

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

> PS- There is an orc behind the oak door.

The door is ajar. Place the orc in the jar.

If you do go to

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

note that 4:8:11:13 will give you octaves, which is more likely
something you might want than what I originally suggested.

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Igs:
>
> > Carl:
> > > This won't be badness then. Badness is error * complexity
> > > for a temperament. Complexity means "number of notes".
> > > For an EDO, the EDO number is a good kind of complexity.
> >
> > Okay, I take back my stipulations, then.
>
> Don't do that! I'm just trying to figure out what you want.
> I thought you wanted to *compare* EDOs, in which case badness
> is a good bet. If you want to find resources in each EDO,
> weighted error is a good bet.

I want to compare the best resources in each EDO, but I'm looking at "best" in a different way than I think you are. I'm not looking at concordance. I'm defining "best" as "the chord that that EDO does with less badness than any other EDO". My instinct tells me that while these chords may not always be the highest in concordance, they will at least be likely to be the most "beatless" in the tuning, or the most "rooted", or the most...something. The most likely to work as a power chord, maybe? I dunno, I'm not firing on all cylinders right now.

To be crystal clear: for the purposes of this exercise, I'm tossing "concordance" to the wind, and ranking everything within the "n-limit" the same. So I'm not scoring 8:11:13 any higher than 4:5:6, or whatever. I'm ONLY looking at "error from JI" (i.e. cents deviation from the harmonic series) AND "number of notes per octave".

> Yes, but you didn't seem to like it.

I didn't think about it enough before responding. Now I like it, I'm just not sure I like the "7-limit" part.

> What do you make of the suggestion I just sent?

It's not quite what I'm looking for, because it doesn't supply badness.

> I'm not sure I understand. Here's what I know how to do:
>
> Find the best chord for each ET
> Find the best ET for each chord
>
> Which path do you choose?

I believe 'tis the latter.

The question is what do you mean "each chord"? Suppose we define the list of chords to evaluate as, "all the chords in some odd-limit." Let's call that odd limit "n". I don't know the value of n right now.

What I suspect is that there is some value for n wherein *any* given EDO from 5 to 37 can be found to give the lowest badness for at least ONE n-limit triad, or at the very least CLOSE to the lowest badness of any EDO in the range. I want to know what this n-limit is.

I have sort of a hunch that the value of "n" might be somewhere around 21 or 23. That covers a lot of triads, anyway, and would be a good starting-place. (Oh, but let's also assume that our triads span less than an octave--that'll cut down on "repeats" and octave-doublings, since we have pure octaves in all tunings). The goal is to have the lowest limit possible such each EDO gives the lowest (or near-lowest) badness for at least ONE triad in the n-limit, so if all EDOs give the lowest badness for MORE THAN 1 triad, the limit may be too high. Maybe it's only 19 or 17? I have no idea. This is why I'm coming to you.

Does this make sense, or am I still failing to articulate myself? At any rate, THANK YOU for bearing with me! If this pans out, I think I'll be satisfied.

-Igs

Igs:

> My instinct tells me that while these chords may not always
> be the highest in concordance, they will at least be likely
> to be the most "beatless" in the tuning, or the most "rooted",
> or the most...something.

When you figure it out, let me know.

> The most likely to work as a power chord, maybe? I dunno,
> I'm not firing on all cylinders right now.

I think we've been over the available options. You can
use one of them, or roll your own.

> To be crystal clear: for the purposes of this exercise,
> I'm tossing "concordance" to the wind, and ranking everything
> within the "n-limit" the same. So I'm not scoring 8:11:13
> any higher than 4:5:6, or whatever.

Whether those rate the same depends on n.

> > Yes, but you didn't seem to like it.
>
> I didn't think about it enough before responding. Now
> I like it, I'm just not sure I like the "7-limit" part.

My latest proposal uses a Tenney limit of 1000. All the
chords where cubert(a*b*c) <= 10. Then you create a
weighted error by multiply their Tenney height by their
RMS error in the given ET. Then you find the best, or best
three, chords for that ET. Then if you really want you
can compare ETs by the score of their best chords.

> What I suspect is that there is some value for n wherein *any*
> given EDO from 5 to 37 can be found to give the lowest badness
> for at least ONE n-limit triad, or at the very least CLOSE to
> the lowest badness of any EDO in the range. I want to know
> what this n-limit is.

Oh hm, if you want to use badness that way. I suppose we
could try it both ways. But I'll use Tenney limit instead
of odd limit, at least for now, because it's easier to
compute.

> Does this make sense, or am I still failing to articulate
> myself? At any rate, THANK YOU for bearing with me! If
> this pans out, I think I'll be satisfied.

Give me 24 hours. -Carl

Igs>"I'm not looking at concordance. I'm defining "best" as "the chord that
that EDO does with less badness than any other EDO"...So I'm not scoring 8:11:13
any higher than 4:5:6, or whatever. I'm ONLY looking at "error from JI" (i.e.
cents deviation from the harmonic series) AND "number of notes per octave"."

In other words...you are defining EDO where it's version of a certain chord
has less error from the "perfect" JI version than all the other EDOs? Then the
person would, say, pick his/her favorite chords and find the EDO that contains
the most of them?

>"What I suspect is that there is some value for n wherein *any* given EDO from 5
>to 37 can be found to give the lowest badness for at least ONE n-limit triad, or
>at the very least CLOSE to the lowest badness of any EDO in the range. I want to
>know what this n-limit is. "
Correct me if I'm wrong, but I think a good starting point would be which TET
approximates which limit dyads the best. And 31TET has a whole bunch...5,7, and
11...so you'd think with a common denominator at those primes would do quite
well in that tuning, for example. Then you'd just take, for example, all the
scales good for, say, 7-limit dyads and see which of those is best for certain
7-limit triads, 4-tone chords, and so on.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> In other words...you are defining EDO where it's version of a certain chord
> has less error from the "perfect" JI version than all the other EDOs? Then the
> person would, say, pick his/her favorite chords and find the EDO that contains
> the most of them?

More or less, yes.

> Correct me if I'm wrong, but I think a good starting point would be which TET
> approximates which limit dyads the best.

Yeah, but then you have to take into account the size of the EDO, too, and I don't really know the proper way to weight it. But I think Carl's on it.

>And 31TET has a whole bunch...5,7, and
> 11...so you'd think with a common denominator at those primes would do quite
> well in that tuning, for example.

Yeah, but it gets tricky because error gets multiplied when you start combining intervals...like 31 has a decent but not great 11, and a decent but not great 15, but lo and behold it's 15/11 is the same interval as its 11/8 (from a strictly error standpoint). And sometimes an EDO will totally nail an interval like 7/6 despite totally lacking either 6 or 7 (see 9-EDO). So while having the prime/odd-factors approximated guarantees that combination intervals will be approximated, NOT having primes doesn't mean that combination intervals WON'T be approximated. It's tricky.

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > My instinct tells me that while these chords may not always
> > be the highest in concordance, they will at least be likely
> > to be the most "beatless" in the tuning, or the most "rooted",
> > or the most...something.
>
> When you figure it out, let me know.

Well, this is my instinct talking. We'll have to see what the experiment produces.

> > The most likely to work as a power chord, maybe? I dunno,
> > I'm not firing on all cylinders right now.
>
> I think we've been over the available options. You can
> use one of them, or roll your own.

Let's see what you can do with the idea you proposed.

> > To be crystal clear: for the purposes of this exercise,
> > I'm tossing "concordance" to the wind, and ranking everything
> > within the "n-limit" the same. So I'm not scoring 8:11:13
> > any higher than 4:5:6, or whatever.
>
> Whether those rate the same depends on n.

Well, yeah, but I'm assuming n>13 at least. I don't think a lot of EDOs will look very good if we don't look above 13-odd-limit.

> My latest proposal uses a Tenney limit of 1000. All the
> chords where cubert(a*b*c) <= 10. Then you create a
> weighted error by multiply their Tenney height by their
> RMS error in the given ET. Then you find the best, or best
> three, chords for that ET. Then if you really want you
> can compare ETs by the score of their best chords.
>
> > What I suspect is that there is some value for n wherein *any*
> > given EDO from 5 to 37 can be found to give the lowest badness
> > for at least ONE n-limit triad, or at the very least CLOSE to
> > the lowest badness of any EDO in the range. I want to know
> > what this n-limit is.
>
> Oh hm, if you want to use badness that way. I suppose we
> could try it both ways. But I'll use Tenney limit instead
> of odd limit, at least for now, because it's easier to
> compute.

Can we use a higher Tenney Height? I know, anything above 1000 is probably discordant, but something like 8:11:13 is still beatless and it's 1144. I want to look at chords at least up to the 21-odd-limit, discordance be damned, because one of my favorite triads is 16:18:21. That comes out to 6048, so is it too much to ask that you take it up at least that far?

> > Does this make sense, or am I still failing to articulate
> > myself? At any rate, THANK YOU for bearing with me! If
> > this pans out, I think I'll be satisfied.
>
> Give me 24 hours.

Gladly! Thanks again, Carl!

-Igs

> Can we use a higher Tenney Height? I know, anything above 1000
> is probably discordant, but something like 8:11:13 is still
> beatless and it's 1144. I want to look at chords at least up to
> the 21-odd-limit, discordance be damned, because one of my
> favorite triads is 16:18:21. That comes out to 6048, so is it
> too much to ask that you take it up at least that far?

It's not that > 1000 is discordant, just that it might be
approximating something < 1000. But n will be a free parameter.
We can make it anything we want. I will also implement the
within-one-octave thing to avoid things like 1:2:499. -C.

Igs>> Can we use a higher Tenney Height? I know, anything above 1000
>> is probably discordant, but something like 8:11:13 is still
>> beatless and it's 1144. I want to look at chords at least up to
>> the 21-odd-limit, discordance be damned, because one of my
>> favorite triads is 16:18:21. That comes out to 6048, so is it
>> too much to ask that you take it up at least that far?

Carl>It's not that > 1000 is discordant, just that it might be
>approximating something < 1000.

Right, so how do we get from > 1000 to < 1000, especially considering 8:11:13
can't be simplified periodically (IE 4:5:7 doesn't exactly work)?
One way I can think of it to multiply the Tenney Height result by the critical
band dissonance of 11/8 plus 13/11. Any others?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> Right, so how do we get from > 1000 to < 1000,

What does this mean?

-Carl

Me>> Right, so how do we get from > 1000 to < 1000,
Carl>What does this mean?
You said that even a chord with an Tenney Height > 1000 could be rounded to
something (I'm assuming you meant another chord) that would have a Tenney height
of under 1000 as part of your solution....correct?
Now, assuming that, how would you do that rounding?

Michael wrote:

>>> Right, so how do we get from > 1000 to < 1000,
>>
>> What does this mean?
>
> You said that even a chord with an Tenney Height > 1000 could
> be rounded to something (I'm assuming you meant another chord)
> that would have a Tenney height of under 1000 as part of your
> solution....correct?

Some chords *approximate* other chords. Some chords of
Tenney height < 1000 will approximate one another. But
more chords of Tenney height > 1000 will do this.

> Now, assuming that, how would you do that rounding?

Not sure what you would like to do... nobody knows
exactly which chords approximate which other chords.
It's uncharted territory.

-Carl

>"Not sure what you would like to do... nobody knows
exactly which chords approximate which other chords."

How to explain this....
Here's the context of my message
Igs> Can we use a higher Tenney Height? I know, anything above 1000
> is probably discordant, but something like 8:11:13 is still
> beatless and it's 1144. I want to look at chords at least up to
> the 21-odd-limit, discordance be damned, because one of my
> favorite triads is 16:18:21. That comes out to 6048, so is it
> too much to ask that you take it up at least that far?

Igs...correct me if I'm wrong, but doesn't that imply you find a problem with
Tenney Height in the case of analyzing the chords you mentioned IE they are
beat-less to you, yet have very high Tenney Heights nonetheless?

Carl (in reply)>"It's not that > 1000 is discordant, just that it might be
approximating something < 1000."

Doesn't that mean that a chord over 1000 in Tenney Height may approximate one
under 1000?

Regardless if that's how you meant it...rounding chords to lower limit
ones (that would naturally give lower Tenney Height) seems like a good idea to
me IF someone can figure out a way to find out "which chords approximate which
other chords" of lower Tenney limit.

And if that can't be done, I was suggesting, for example, that the
critical band dissonance of the two consecutive dyads in a chord come into play
IE for 8:11:13 that we use the critical band dissonance of 11:8 and 13/11 to
help multiple/scale the Tenney Height by. This way the fact 11:8 is fairly low
on critical band dissonance (especially compared to, say, 11/10) I figure will
help accomplish getting the measure more in line with lower-Tenney-Height
chords.

Michael,

You've replied to both Igs and myself in a way that's
confusing to me -- I don't think all the context came
through. But I'll try my best to answer.

> Carl (in reply)>"It's not that > 1000 is discordant, just
> that it might be approximating something < 1000."
>
> Doesn't that mean that a chord over 1000 in Tenney Height
> may approximate one under 1000?

It means exactly what I said in my previous message,
no more, no less.

> Regardless if that's how you meant it...rounding chords to
> lower limit ones (that would naturally give lower Tenney Height)
> seems like a good idea to me IF someone can figure out a way
> to find out "which chords approximate which other chords" of
> lower Tenney limit.

What is "rounding" a chord?

-Carl
> And if that can't be done, I was suggesting, for example, that the
> critical band dissonance of the two consecutive dyads in a chord come into play
> IE for 8:11:13 that we use the critical band dissonance of 11:8 and 13/11 to
> help multiple/scale the Tenney Height by. This way the fact 11:8 is fairly low
> on critical band dissonance (especially compared to, say, 11/10) I figure will
> help accomplish getting the measure more in line with lower-Tenney-Height
> chords.
>

Michael,

Sorry, missed this bit. Are you using the rich text editor
to compose your messages? They are uniquely frustrating to
read and reply to.

> And if that can't be done, I was suggesting, for example,
> that the critical band dissonance of the two consecutive
> dyads in a chord come into play IE for 8:11:13 that we use
> the critical band dissonance of 11:8 and 13/11 to help
> multiple/scale the Tenney Height by. This way the fact 11:8
> is fairly low on critical band dissonance (especially
> compared to, say, 11/10) I figure will help accomplish
> getting the measure more in line with lower-Tenney-Height
> chords.

Sounds like a good idea - why don't you do it?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
In reply to Michael, Carl asked:
> What is "rounding" a chord?

I presume he means the process of reducing a high-limit (or large Tenney Height) chord to whatever lower limit/Tenney Height you'd say it's approximating. I.e. how you could "round" 16:19:24 to 10:12:15, or something.

-Igs

>"What is "rounding" a chord?"
For example

100:126:167 can round to approximately 5:6:8 since
A) 6/5 is close to 126/100 and
B) 167/100 is close to 5/3...and 5/3 * 5 = about 8

you had said,
> Carl (in reply)"It's not that > 1000 is discordant, just
> that it might be approximating something < 1000.
...and that does seem to indirectly imply that > 1000 is likely not
discordant because, to the brain it's "really approximated by the brain to be
less than 1000" IE via the kind of rounding that Harmonic Entropy entails.

Now what you seem to be saying is you meant "not only is > 1000 not
discordant, but it may be so without any regards to the brain rounding it to a
lower limit chord (AKA a chord with a likely lower Tenney Height)".

With all this floating around, the greater question becomes how do may any of
these opinions (mis-interpreted or not) help us solve what appears to be the
issue Igs found: that some chords with high Tenney Heights are relatively
beatless and, if I have Igs's opinion right, surprisingly conSONant sounding
despite having such high Tenney Heights?

Carl>"Sounds like a good idea - why don't you do it?"

Ok, let me work this one out...note I'm using a visual graph on
http://eceserv0.ece.wisc.edu/%7Esethares/images/image3.gif
as a reference so results might be a decimal point or so off.

13/11 = 1 critical band dissonance
11/8 = 0.125 critical band dissonance
2.00 = maximum on Sethares' graph (at about the 12TET semitone)
3.00 = number of tones in chord
6.00 maximum "sequential dyad" critical band dissonance for 3 note chord.
1144 (Igs's original Tenney height of 8*11*13)

So for 8:11:13...
((1 + 0.125) / (2.00*3.00)) * 1144 = 214.5 critical band-scaled Tenney Height
(about 1/5th the original)

Now the bad news is something like Igs suggestion of 16:18:21 (Tenney height
calculated the way Igs did for that is 6048) isn't going to be reduced that
much. Say
18/16 = 9/8 = 1.4 critical band dissonance
21/18 = 7/6 = 0.4 critical band dissonance
6.00 maximum "sequential dyad" critical band dissonance for 3 note chord.
6048 Tenney Height

((1.4 + 0.4) / (2.00*3.00)) * 6048 = 1814.4 critical band-scaled Tenney Height
(about 1/3rd the original Tenney Height)

So it does seems to work in the case of bringing down the Tenney Height more
of chords which are less closely spaced (IE 8:11:13 is less closely space than
16:18:21 and thus gets better reduction) and gives good credit to chords that
have fair to good spacing. However it still can't save Igs's 16:18:21 because
the original Tenney Height is so unbearably high that the reduction algorithm
can't lower it despite it's having not-so-bad critical band values.
-----------------------------------

To alleviate that problem, might I suggest calculating Tenney Height using
square roots (IE the method Carl first mentioned) and then changing the "limit
of consonance" from 1000 to something much lower? I'd do it myself, but I'm
not 100% sure how the math for it works....

________________________________
From: Carl Lumma <carl@...>
To: tuning@yahoogroups.com
Sent: Thu, September 9, 2010 5:28:31 PM
Subject: [tuning] Re: n-Limit Triads & Various EDOs

Michael,

Sorry, missed this bit. Are you using the rich text editor
to compose your messages? They are uniquely frustrating to
read and reply to.

> And if that can't be done, I was suggesting, for example,
> that the critical band dissonance of the two consecutive
> dyads in a chord come into play IE for 8:11:13 that we use
> the critical band dissonance of 11:8 and 13/11 to help
> multiple/scale the Tenney Height by. This way the fact 11:8
> is fairly low on critical band dissonance (especially
> compared to, say, 11/10) I figure will help accomplish
> getting the measure more in line with lower-Tenney-Height
> chords.

Sounds like a good idea - why don't you do it?

-Carl

Michael wrote:

> Ok, let me work this one out...note I'm using a visual graph on
> http://eceserv0.ece.wisc.edu/%7Esethares/images/image3.gif
> as a reference so results might be a decimal point or so off.
>
> 13/11 = 1 critical band dissonance
> 11/8 = 0.125 critical band dissonance
> 2.00 = maximum on Sethares' graph (at about the 12TET semitone)
> 3.00 = number of tones in chord
> 6.00 maximum "sequential dyad" critical band dissonance
> for 3 note chord.
> 1144 (Igs's original Tenney height of 8*11*13)

The graph shows something resembling Plomp-Levelt purity
for two sine tones. Of course knowing the units on the
graph is an important first step... whatever they are, 11/8
seems to have more like half of what 13/11 has than a tenth.
Not sure why you're giving number of notes in a chord to two
decimal places, since it must be a whole number...

> So for 8:11:13...
> ((1 + 0.125) / (2.00*3.00)) * 1144 = 214.5 critical
> band-scaled Tenney Height (about 1/5th the original)

What about 13/8?

> Now the bad news is something like Igs suggestion of
> 16:18:21 (Tenney height calculated the way Igs did for
> that is 6048) isn't going to be reduced that much. Say
> 18/16 = 9/8 = 1.4 critical band dissonance
> 21/18 = 7/6 = 0.4 critical band dissonance
> 6.00 maximum "sequential dyad" critical band dissonance
> for 3 note chord.
> 6048 Tenney Height
>
> ((1.4 + 0.4) / (2.00*3.00)) * 6048 = 1814.4 critical band-scaled
> Tenney Height (about 1/3rd the original Tenney Height)

Why are you assuming sine tones? Don't you want sensory
dissonance for a generic harmonic timbre?

> To alleviate that problem, might I suggest calculating
> Tenney Height using square roots (IE the method Carl first
> mentioned)

Cube roots in this case.

-Carl

Hi Carl, any progress on this yet?

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > Can we use a higher Tenney Height? I know, anything above 1000
> > is probably discordant, but something like 8:11:13 is still
> > beatless and it's 1144. I want to look at chords at least up to
> > the 21-odd-limit, discordance be damned, because one of my
> > favorite triads is 16:18:21. That comes out to 6048, so is it
> > too much to ask that you take it up at least that far?
>
> It's not that > 1000 is discordant, just that it might be
> approximating something < 1000. But n will be a free parameter.
> We can make it anything we want. I will also implement the
> within-one-octave thing to avoid things like 1:2:499. -C.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi Carl, any progress on this yet?
>
> -Igs

I sent you a message offlist - did you get it? -C.

Carl>"Not sure why you're giving number of notes in a chord to two
decimal places, since it must be a whole number..."
I'm just doing that to be consistent with the other values for consonance.

>"Of course knowing the units on the graph is an important first step"
Right, and these aren't given. Knowing an exact equation that estimates the
graph (regresssion?) would help as well. Do you happen to have one?

>"11/8 seems to have more like half of what 13/11 has than a tenth."
Well, it's hard to read such a small graph without line tickings on the
side...(your answer) could very well be.

>> So for 8:11:13...
>> ((1 + 0.125) / (2.00*3.00)) * 1144 = 214.5 critical
>> band-scaled Tenney Height (about 1/5th the original)
>What about 13/8?
I didn't do all dyads in the chord since the third dyad in most chords is
spread out enough that it has little effect and root-tone critical band
dissonance and doesn't differ much between chords (IE the critical band
dissonance difference between 5/3 and 3/2 on the above graph, for example, is
negligible).

>"Why are you assuming sine tones? Don't you want sensory dissonance for a
>generic harmonic timbre?"
In this case, no. I'm assuming Tenney Height itself takes care of things
like overtone alignment (isn't that fair to say: that fractions that align less
well along the harmonic series will generally have much higher Tenney
Heights)...and purposefully trying to avoid factoring in influences of harmonic
alignment twice (which would make the equation again rather biased toward
harmonic alignment and harmonic "complexity").
So I'm using the sine tone graph because I am, on purpose, looking for root
tone dissonance and root tone dissonance ONLY for the scaling.

>"Cube roots in this case."
Ok, such as cube root of 1044 for the original Tenney Height of 1044 where
it's cube root, not square root...as the chord has three, and not three ,
notes. I will also try your new critical band dissonance value for 13/11 of 0.5
IE half of 1.0. And yes, I am sticking to sine tones for critical band
calculation the reasons explained above IE P&L's graph rather than Sethares'
"Harmonic Timbre" one.

Michael wrote:

> >Of course knowing the units on the graph is an important first
> >step
>
> Right, and these aren't given. Knowing an exact equation that
> estimates the graph (regresssion?) would help as well. Do you
> happen to have one?

Uh... Sethares has published the code.

>> What about 13/8?
> I didn't do all dyads in the chord since the third dyad in
> most chords is spread out enough that it has little effect

All dyads in the chord have an effect.

>> Why are you assuming sine tones? Don't you want sensory
>> dissonance for a generic harmonic timbre?
> In this case, no. I'm assuming Tenney Height itself
> takes care of things like overtone alignment (isn't that
> fair to say: that fractions that align less well along the
> harmonic series will generally have much higher Tenney
> Heights)...and purposefully trying to avoid factoring in
> influences of harmonic alignment twice

For dyads of harmonic timbres, the agreement between any
'simple ratios' rule like Tenney height and roughness will be
good. For triads, less good. Anyway it seems to me if
you're going to weight for roughness you should assume a
typical timbre, not an atypical one (sines). But this is
your choice.

> > Cube roots in this case.
> Ok, such as cube root of 1044 for the original Tenney Height
> of 1044 where it's cube root, not square root...as the chord has
> three, and not three , notes.

Yes, it has three , and not three notes. :)

> I will also try your new critical band dissonance value for
> 13/11 of 0.5 IE half of 1.0.

The next thing to try is computing actual sensory
dissonance values for all the dyads in the chord.

-Carl

>"Uh... Sethares has published the code."
Isn't that for critical band dissonance involving a timbre as input?
I'm looking for a formula for P&L's sine wave curve, not Sethares timbre-based
"sum of all partial dissonance" curve. If P&L's curve can be extracted through
parts of Sethares', it would help to know how.

>> What about 13/8?
> I didn't do all dyads in the chord since the third dyad in
> most chords is spread out enough that it has little effect

>All dyads in the chord have an effect.
I said "has little effect" -Me for wider ones, no NO effect. :-S If you
really think it's necessary though, you can throw it in there...it can't hurt,
it will likely give you a tiny bit more accuracy. Would start to mean a lot
more calculation to be done on larger chords, though.

>"For dyads of harmonic timbres, the agreement between any 'simple ratios' rule
>like Tenney height and roughness will be good. For triads, less good."
My assumption is that root-tone (not harmonic) critical band dissonance
would favor wide spacing the most, which is the key advantage the chords Igs
mentioned have or lower limit but often closer spaced ones.
The entire reason I'm using roughness is to counter some of the effects of
Tenney Height that overly favor low-limit chords, not increase them! :-P

>"Yes, it has three , and not three notes. :)"
Blah, a slip on my part, 3 and not two notes implies cube and not square
root.

>> I will also try your new critical band dissonance value for
>> 13/11 of 0.5 IE half of 1.0.
>The next thing to try is computing actual sensory
>dissonance values for all the dyads in the chord.
Ok ok I'll at least try that when I'm done with everything else...
But I don't think it's going to help the balance much and it will involve
an extra calculation for triads, an extra three calculations for tetrads, etc.

Michael wrote:

> >"Uh... Sethares has published the code."
>
> Isn't that for critical band dissonance involving a timbre
> as input?

Yes but I assume you can use sines for the timbre.

> I'm looking for a formula for P&L's sine wave curve,

Sethares gives one in his paper.

> > All dyads in the chord have an effect.
>
> I said "has little effect" -Me for wider ones, no NO effect.

If you're just using P&L / assuming sines, then it will
have little effect. But it doesn't hurt to include it
anyway. In fact, it helps.

> >The next thing to try is computing actual sensory
> >dissonance values for all the dyads in the chord.
> Ok ok I'll at least try that when I'm done with
> everything else...

How can you do anything else before you've done this?
Wait- don't answer that.

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > Those are the two principle versions. Beyond that, it's
> > mainly how much noise you want to assume (the variable "s")
> > and whether you want to take the exponential when you're
> > done. Goldstein deduced that s was about 1%, but Paul tried
> > values between 0.6% and 1.2%. Making it smaller causes more
> > local minima to appear.
>
> So decreasing "s" increase the number of local minima, but otherwise does not alter the distribution around the minima, or the depth of the minima?

I have now completed my code for H.E. and obtained curves that look about right, although my values are not identical to those Carl sent me some time ago. In the range 0-2400 cents my graph has 19 minima (excluding the endpoints) for s=0.6%, but 45 minima for s=1.2%. The depth of the minima is greater in the latter case.

Steve M.

>> >"Uh... Sethares has published the code."
>>
>> Isn't that for critical band dissonance involving a timbre
>> as input?
>Yes but I assume you can use sines for the timbre.
Good idea...that would simplify the result for this special case
accordingly. Got it. Now I just have to rip apart all those inner-locking
loops in his C code and figure out how to map a fraction to the first input
array.

>> I'm looking for a formula for P&L's sine wave curve,
>Sethares gives one in his paper.
I saw e^(-3.5x)-e^(-5.75x) in his paper (x is the difference in frequencies
of the two sine waves involved). But when I actually plugged it into a program
I got oddly scaled answers (IE very tiny numbers, certainly not 0 to 1 or
anything I could use directly).
Is there some factor there I'm missing? I did mail him about it.

it would be nice to map this curve directly...but if I can't get a direct
answer to that I suppose I could use the slow and inefficient technique of
looping through summing the dissonance of all overtone dyads in the entire
timbres formed over the two root notes (what Sethares' full algorithm does) and
setting those overtones to amplitude zero.

>"If you're just using P&L / assuming sines, then it will have little effect.
>But it doesn't hurt to include it
anyway. In fact, it helps."
True enough. I'll try and sneak it in once everything else works.

> >The next thing to try is computing actual sensory
> >dissonance values for all the dyads in the chord.
> Ok ok I'll at least try that when I'm done with
> everything else...
>How can you do anything else before you've done this?
I'm doing some of the dyads (the closest ones that differ most in
consonance first) to get an idea if the scaling is "equalizing" the Tenney
Height values well. I figure more variables and consonance results to manage
could likely mean if there's a problem I have less chance of trimming it down to
one variable and fixing it quickly.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> I saw e^(-3.5x)-e^(-5.75x) in his paper (x is the difference
> in frequencies of the two sine waves involved). But when I
> actually plugged it into a program I got oddly scaled answers
> (IE very tiny numbers, certainly not 0 to 1 or anything I could
> use directly).

Yep, that's the formula (if you're ignoring absolute frequency
and amplitude). Looks sensible to me

http://bit.ly/dhHuhb

Who cares how small the numbers are? Scale them up by whatever
factor you like.

-Carl

Carl>"Yep, that's the formula (if you're ignoring absolute frequency
and amplitude). Looks sensible to me
http://bit.ly/dhHuhb
Who cares how small the numbers are? Scale them up by whatever
factor you like."

Right, but I need to know the lowest and highest possible value to figure out
the top and bottom of the scale.
Plus, you're right....knowing the starting root tones/"absolute frequencies"
would help.
It begs the question (yes this is to everyone): what chords would you rate as
the most and least dissonant that would fit into the category of acceptable
dissonance?

Ideally it would be nice if I could scale it so the "worst tolerable" chords
far as dissonance would have a value of 100 from the formula and the "best"
chords would near zero. That way, I figure, you could easily see what
percentage each chord was from minimum or maximum "tolerable" dissonance.