Yahoo Tuning Groups Ultimate Backup tuning Mike B.'s Hypothesis of Minorness

On Sun, Sep 19, 2010 at 6:50 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Mike, since you've elucidated your hypothesis about minorness primarily through examples, rather than formal statements, let me take a stab at formalizing it.

I've tried to formalize it, but it didn't seem to click with anyone...
:) I'm simply noticing a pattern and have a half-assed hypothesis,
which you are free to make your own judgements about. The strength of
the pattern speaks for itself.

> It appears that your hypothesis is based on two things: the harmonic series, and a "depriming interval" which is the absolute maximum of Harmonic Entropy, around 60-70 cents--perhaps around the size of a 5-limit chromatic semitone (25/24)? Basically, what you hypothesize is that if you start with a pure harmonic series chord of between 3 and--what, 7 or 8 notes?--the starting chord will be "major", but if you then start "detuning" harmonics by the amount of the depriming interval, the chords move away from sounding "major" and toward sounding "negative/painful".

The strong form of my hypothesis is this, yes, with this "depriming"
interval taking a role. I'm not entirely sure that it's right, and
have now modified it a bit to just think that it's a good rule of
thumb to find maximally discordant intervals. This is just a
conjecture, and it's a side thing. Don't get so hung up on this that
the main point is missed.

> "Minorness", then, is sort of a point intermediately between "major" and "painful"?

Yes. This is, in essence, the hypothesis, irrespective of any
particular psychoacoustic explanation. We've been using this in common
practice music for hundreds of years: minor is sadder than major, and
diminished is sadder than minor. And with minor, minor7 is a bit less
sad, and minor maj7 is more so. Chords like C-E-F#-B-D# in 12-tet and
C-Eb-G are on some roughly similar level of "pain." Or C-E-G-B-D-F#,
perhaps. And if you continue the pattern up the harmonic series, it
holds and leads to new xenharmonic "minor" chords that to my ears,
continue the "minor" pattern more than the otonal/utonal thing does.
Note that utonalness has no specific psychoacoustic explanation, and
coincides with detunings of the harmonic series.

> I really don't get how this theory deals with chord formed by higher chunks of the harmonic series. Are you suggesting that chords like 10:12:15 or 6:7:9 are really heard as just "deprimed" versions of 4:5:6? I'd think you sort of *have to* say this for it to make sense, or else 6:7:9 should sound "major", and then its "deprimed" version (which would be close to 8:9:12) should sound more minor.

I'm saying that if you actively TRY to find the most discordant
sonority that emerges when you sharpen the 5 in 4:5:6, it ends up
being somewhere near 10:12:15, or maybe 0-333-702. Go past this and it
gets less discordant. Discordance corresponds to perceptual "pain." If
you try to sharpen the 5, the most discordant sonority emerges
somewhere around where the third is 7:9, and gets less discordant at
10:13:15. Note that 10:13:15 and 6:7:9 are of a roughly similar
quality, and that 0-333-702 and 14:18:21 are as well.

> I'll be damned if I can hear a sus2 chord as "more minor" than a subminor chord. So how exactly *does* your theory deal with triads formed of higher harmonics, or triads where the "deprimed" version is still close to the harmonic series?

You probably don't hear subminor as more painful than minor either, or
a supermajor chord where the major third is 13/10 as more painful than
one where it's 9/7. Note that these also correspond to lower integer
ratios; 6:7:9 is lower than 10:12:15, and 10:13:15 is lower than
14:18:21. Moving an interval by 60-70 cents is a "rough guideline" to
find intervals that are maximally discordant from these. The "truest"
minor chord, in a sense, is one in which you screw with that minor
third to get it to sound as "biting" as possible. Then you can adjust
it, if you want, to something like 6/5, so it's beatless.

Perhaps a better generalization would be the "Noble Mediant" approach,
but generalized somehow to harmonies without dyads.

That's why I modified the hypothesis. The "depriming" interval is just
sort of a rule of thumb to find maximally discordant intervals nearby.

Nonetheless, if you feel there is a better explanation, please give
it. Maybe it just boils down to the harmonic entropy of a tetrad.
However, not only does the pattern work, and not only does it explain
all of the 12-tet chords I've ever played, but it's giving me a
fascinating new perspective with which to view 13-tet, and a way to
generalize tonality to other sections of the overtone series which I
think will yield good results. If you don't hear it the same way, then
alright... I'll view that as an interesting data point and assume that
perhaps I've just found a general pattern I've "learned" to hear as
sad or painful. So far, most of my friends have heard the "series of
negativity" in my example as increasingly negative though, so I hope
the lack of interest here is just in that my initial psychoacoustic
conjecture put people off in being perhaps somewhat simplistic.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Sun, Sep 19, 2010 at 8:49 PM, Mike Battaglia <battaglia01@...> wrote:
> On Sun, Sep 19, 2010 at 6:50 PM, cityoftheasleep
>> It appears that your hypothesis is based on two things: the harmonic series, and a "depriming interval" which is the absolute maximum of Harmonic Entropy, around 60-70 cents--perhaps around the size of a 5-limit chromatic semitone (25/24)? Basically, what you hypothesize is that if you start with a pure harmonic series chord of between 3 and--what, 7 or 8 notes?--the starting chord will be "major", but if you then start "detuning" harmonics by the amount of the depriming interval, the chords move away from sounding "major" and toward sounding "negative/painful".

One last thing: another good reason to use something like 60 cents as
the "standard interval of adjustment" is that it leads to an intuitive
extension of regular tempering. You have a new "prime," which we'll
call #. You can then "temper" that with other nearby JI ratios, like
25/24 for meantone, or 16/15 for superpyth.

So minor would be 4:5b:6. You could temper 4:5b to be equal to 5:6,
and then it would become equal to 10:12:15. Perhaps this could be
called "minor" temperament or something.

This might lead to linear temperaments with generators we wouldn't
really care about otherwise, but are notable for producing # with
enough complexity that it doesn't dominate, but exists in a diatonic
scale.

But either way, there are really ever-increasing stranger hypotheses
that emerge from this "detuning = pain" principle, which I think is
somewhat simple and intuitive, which I'll reveal as soon as I've sold
you on the first part. I don't want to alienate anyone else with
things that just sound like pseudopsychoacoustics at this point :)

The last few days have really seen an incredible transformation in the
way I perceive music though... Things like 13-tet have really popped
to life. Detuned JI sonorities are no longer "unusable," but can be
seen as causing slight amounts of "pain..." this can be manipulated to
create musical feeling to an astonishingly predictable extent. The
space between minor and painfulness is what you seem to be in
agreement on... the space between minorness and majorness I find far
more interesting, however.

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Sun, Sep 19, 2010 at 11:08 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > I'm sorry, I may have perceived this as more of an attack than it was.
> > Are you saying that it's not likely to be able to formalize this as
> > part of regular mapping?
>
> I don't think it's possible that a particular and precisely defined transcendental number can emerge from your very general considerations. You need something more, whether noble mediants or something else.

I think that something like 60 cents might be useful to start with,
but I can see your desire to make it more mathematically rigorous. How
about this:

If I define two operators b and #, such that

4:5b:6 is an "idealized" minor chord, in which 4:5b:6 ends up being
maximally discordant from all nearby otonal triads, and
4:5#:6 is an "idealized" supermajor chord, in which 4:5#:6 ends up
being maximally discordant from all nearby otonal triads

Then do you think that something more rigorous could be formalized? So
there would be an additional set of "maximally discordant" intervals
that would be generated from the existing set of concordant ones.

Here's how such a function could be defined, and what I'm currently working on:

Harmonic entropy first matches an incoming to the set of all dyads,
generated with a Tenney series, then makes an entropy calculation. A
more elegant way to do this, perhaps, would be to come up with a
transform that treats the dyad or triad or any-ad as a signal, and
represents it with a set of non-orthogonal basis functions (with the
Fourier transform as an orthogonal subset) which have the frequency
response of a damped sinusoid with its frequency as a parameter, its
damping coefficient as a parameter, AND a set of harmonics with
rolloff as third parameter. If you get rid of the last argument, you
have the Laplace transform (if you use complex exponentials instead of
sinusoids). So the addition of the last argument would, effectively,
extend the Laplace transform into a three-dimensional space, in which
the additional dimension represents the presence of harmonics with a
rolloff that increases along each side of the axis. Seems I should
call it the "Harmonic Laplace Transform," which is a name I just made
up, so HLT from now on. There's probably a clever way to get the
computation of this down to nlog(n) time or something.

So now I pick some rolloff r that corresponds to the human
neurological auditory filter, and a damping coefficient d that spreads
the cone of uncertainty of each harmonic to about 300 Hz (the width of
a critical band), and hence we will obtain the set of all basis
vectors that "match" the signal, and how much. Use this as the basis
for an entropy calculation (for any amount of notes, with any timbre,
for a time-varying signal if you want), and profit. Then, the "#" and
"b" operators in JI can be rigorously defined as the local maximum of
entropy nearby the detuning of a note sharp or flat, either with sines
or for some standard "reference timbre" if we prefer.

That's a bit more mathematically rigorous; hopefully you can see how
this fits into the abstract topology that you have going on with
regular temperament. Although you're a math Ph.D, I don't know how
much you had to delve into the specifics of things like the Laplace
transform (I remember you asking some questions about it), so I can go
more into the exact formulation of the transform and how I propose to
extend it, if you want.

-Mike

🔗Mike Battaglia <battaglia01@...>

🔗cityoftheasleep <igliashon@...>

Hi Mike,

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > "Minorness", then, is sort of a point intermediately between "major" and "painful"?
>
> Yes. This is, in essence, the hypothesis, irrespective of any
> particular psychoacoustic explanation. We've been using this in common
> practice music for hundreds of years: minor is sadder than major, and
> diminished is sadder than minor. And with minor, minor7 is a bit less
> sad, and minor maj7 is more so. Chords like C-E-F#-B-D# in 12-tet and
> C-Eb-G are on some roughly similar level of "pain." Or C-E-G-B-D-F#,
> perhaps. And if you continue the pattern up the harmonic series, it
> holds and leads to new xenharmonic "minor" chords that to my ears,
> continue the "minor" pattern more than the otonal/utonal thing does.
> Note that utonalness has no specific psychoacoustic explanation, and
> coincides with detunings of the harmonic series.

So then "major" should be painless? If that's the case, then I think it's not right to call a 4:5:6:7:9:11:13 chord "major", because that mother is painful as F*** to my ears! Though I guess as far as heptads made up of unique notes go, it's probably as painless as you can get. Still, if my experience of that chord suggests anything, it's that harmony gets more painful the more complex it is, so by your theory a 4:5:6 chord would be more major than a 4:5:6:7:9:11:13 chord. I can definitely agree to that.

> I'm saying that if you actively TRY to find the most discordant
> sonority that emerges when you sharpen the 5 in 4:5:6, it ends up
> being somewhere near 10:12:15, or maybe 0-333-702. Go past this and it
> gets less discordant. Discordance corresponds to perceptual "pain." If
> you try to sharpen the 5, the most discordant sonority emerges
> somewhere around where the third is 7:9, and gets less discordant at
> 10:13:15. Note that 10:13:15 and 6:7:9 are of a roughly similar
> quality, and that 0-333-702 and 14:18:21 are as well.

After doing some listening tests, where I used sawtooth, piano, and/or guitar patches, I can conclusively say that 10:13:15 is more discordant to my ears than 14:18:21. I am powerless to explain why this might be, except to hypothesize that it's because the component dyads are simpler in the 14:18:21. 10:13:15 sounds less "major" to me. But from your hypothesis, this makes sense, because 387+60=447, which is closer to 10:13 than it is to 7:9. However, I can also say that if you subtract 60 cents from 387, you get 327, whereas I hear the maximum of discordance closer to 347. 347 hurts, man. It practically draws blood compared to 387, but 333 or 327 just gives a slight bruise. Unsurprisingly, this is in line with H.E.

> You probably don't hear subminor as more painful than minor either, or
> a supermajor chord where the major third is 13/10 as more painful than
> one where it's 9/7. Note that these also correspond to lower integer
> ratios; 6:7:9 is lower than 10:12:15, and 10:13:15 is lower than
> 14:18:21. Moving an interval by 60-70 cents is a "rough guideline" to
> find intervals that are maximally discordant from these. The "truest"
> minor chord, in a sense, is one in which you screw with that minor
> third to get it to sound as "biting" as possible. Then you can adjust
> it, if you want, to something like 6/5, so it's beatless.

Well, you're wrong about my perception of the 13/10 vs. 9/7, but right about the subminor. 6:7:9 is decidedly less painful than 10:12:15 or 14:17:21. But it's also less painful than 26:30:39, which has a 15/13 subminor third. So in some ways, the dyadic H.E. curve holds, but in other ways, it doesn't, since the chord with the 6/5 is more painful than the chord with the 7/6.

> Perhaps a better generalization would be the "Noble Mediant" approach,
> but generalized somehow to harmonies without dyads.
>
> That's why I modified the hypothesis. The "depriming" interval is just
> sort of a rule of thumb to find maximally discordant intervals nearby.

If you used noble mediants, wouldn't the depriming interval vary depending on which two dyads you were looking at? And wouldn't the results come out looking just like H.E.?

> Nonetheless, if you feel there is a better explanation, please give
> it. Maybe it just boils down to the harmonic entropy of a tetrad.
> However, not only does the pattern work, and not only does it explain
> all of the 12-tet chords I've ever played, but it's giving me a
> fascinating new perspective with which to view 13-tet, and a way to
> generalize tonality to other sections of the overtone series which I
> think will yield good results. If you don't hear it the same way, then
> alright... I'll view that as an interesting data point and assume that
> perhaps I've just found a general pattern I've "learned" to hear as
> sad or painful. So far, most of my friends have heard the "series of
> negativity" in my example as increasingly negative though, so I hope
> the lack of interest here is just in that my initial psychoacoustic
> conjecture put people off in being perhaps somewhat simplistic.

Maybe you should stick to comparing triads for now, since (IMHO) adding more unique notes to a chord always increases the pain. Do these predictions hold for triads that span intervals other than a 3/2? What about an 8:9:12?

At any rate, I'm at a loss to come up with a real hypothesis about feelings of minorness. 6:7:9 blows the "otonal=major" theory, 8:9:12 blows the "lower than 5/4=minor" theory, since I swear on my grandmother's grave that 8:9:12 sounds major to me. 9/8 is lower in pitch than 6/5, higher in pain than 7/6, and 8:9:12 has, to my knowledge, lower tonalness than 6:7:9. I also wanted to believe that, given two triads with the same span, the one with the lower middle note will sound minor in comparison to the one with the higher middle note, but comparing 16:18:21 with its utonal counterpart--1:7/6:21/16--kills that. I'll try to post an audio example tomorrow some time, but to me it's pretty obvious.

🔗Mike Battaglia <battaglia01@...>

On Mon, Sep 20, 2010 at 2:07 AM, cityoftheasleep
<igliashon@...> wrote:
>
> So then "major" should be painless? If that's the case, then I think it's not right to call a 4:5:6:7:9:11:13 chord "major", because that mother is painful as F*** to my ears! Though I guess as far as heptads made up of unique notes go, it's probably as painless as you can get. Still, if my experience of that chord suggests anything, it's that harmony gets more painful the more complex it is, so by your theory a 4:5:6 chord would be more major than a 4:5:6:7:9:11:13 chord. I can definitely agree to that.

Is it as painful if the 5:6:7:9:11:13 part of it slowly decreases in
volume as the harmonics get higher? I would imagine not.

I never compared 4:5:6 to 4:5:6:7 or anything like that, nor did I try
to generalize "majorness." I just said that discordance causes
displeasure, and that minorness is a watered down version of that. And
also that a series of progressively more "negative" sounding,
discordant chords can be built from taking successive overtones and
detuning them to areas of maximum discordance, and that some of these
chords have been used in common practice music for centuries. In fact,
I'm not saying anything new; we've already known that diminished
chords sound so dark because the flattened fifth is a discordant and
less stable sonority than the perfect fifth. I simply generalized that
all up and down the harmonic series, and found that when you alter the
5 you get minor, when you alter 5 and 6 you get diminished, when you
alter 5 and 7 and leave six alone, you get min/maj7 or min/maj6 (with
a super sharp six), and that the pattern continues all the way up the
overtone series to new xenharmonic ultra-minor chords. I think that
this is significant, and although I note much has been made of the
fact that 4:5:6:7 implies an even stronger VF than 4:5:6, I'm not
aware of much research on how small amounts of dissonance, introduced
predictably into microtonal sonorities, influences the result.

I also noticed that sharpening the major third in small doses does the
same thing, and can create sort of a feeling of sadness too, unless
it's done too much, at which point it sounds awful, but then starts to
become harmonic again as the 5/4 becomes 4/3. Hence you have maj9#11
chords as sounding kind of "sad," and supermaj9#11 chords sounding
slightly more "sad."

> After doing some listening tests, where I used sawtooth, piano, and/or guitar patches, I can conclusively say that 10:13:15 is more discordant to my ears than 14:18:21. I am powerless to explain why this might be, except to hypothesize that it's because the component dyads are simpler in the 14:18:21.

I really don't hear it that way, but alright. Perhaps there's some
listener-dependent facet of it. I'd be curious to hear how people
compare them in a listening test. I can relax and hear 10:13:15 as
more chilled out after the initial attack. But 14:18:21 bothers me
more.

> 10:13:15 sounds less "major" to me. But from your hypothesis, this makes sense, because 387+60=447, which is closer to 10:13 than it is to 7:9.

I'm not sure how you are abstractly defining "major." I hear 9/7 as
fitting better into the "Major" sound than 13/10, which is a bit
further out. But 9/7 just bothers the hell out of me, and 13/10
doesn't (assuming 3/2 is there as well in both cases).

I don't think that concordance or tonalness leads directly to
"happiness." I think that minorness, however, leads to sadness,
because discordance leads to perceptual pain, and the removal of the
intrusive stimulus and its replacement by the somewhat cool sound of
notes merging and fusing together can lead to some sort of "happy"
feeling after the shrapnel in your ears that is the 13-limit
"negative" sonority I played in the listening example is removed.

I think that part of the "major" sound that we're used to in 12-tet
comes from its approximation to 4:5:6, some comes from the major third
which is slightly sharp and hence a bit "exciting," and part coming
from how we're used to it being a stable tonic in the sea of
dissonance and consonance that is the diatonic scale. But I haven't
really tried to generalize majorness, I'm more on minorness for the
moment.

I want to reiterate, again, that I no longer think that the 60 cents
is a hard and fast rule. It was just a nice rule of thumb. See my
recent letter to Gene for what I really think, if you want to delve
further into the math. I think it boils down to triadic entropy not
being dyadic entropy, and I'm working on a transform to calculate it
for any amount of notes to just put the matter to rest.

> However, I can also say that if you subtract 60 cents from 387, you get 327, whereas I hear the maximum of discordance closer to 347. 347 hurts, man. It practically draws blood compared to 387, but 333 or 327 just gives a slight bruise. Unsurprisingly, this is in line with H.E.

Is this with triads or bare dyads?

But either way, right, now you're getting what I'm trying to
communicate :) Play around and find the most painful sonorities you
can. Forget I said anything about 60 cents. Assuming what you just
said was a triad with 3/2 on top of it, you have something like
4:4.9:6. Now add 7 9 and 11 on top, so you have 4:4.9:6:7:9:11. Lower
the 7, 9, and 11 slightly if you want, so as to avoid them poking out
and being painful. Most importantly, note the feeling that is produced
as you mix concordance and discordance. Then bring the 4.9 back to 5,
and see how it changes. Do this in scala with the "reed organ" patch
if you want to have the same reference sound that I have.

> Well, you're wrong about my perception of the 13/10 vs. 9/7, but right about the subminor. 6:7:9 is decidedly less painful than 10:12:15 or 14:17:21. But it's also less painful than 26:30:39, which has a 15/13 subminor third. So in some ways, the dyadic H.E. curve holds, but in other ways, it doesn't, since the chord with the 6/5 is more painful than the chord with the 7/6.

In many ways, dyads matter, and in many ways, triads matter...

> If you used noble mediants, wouldn't the depriming interval vary depending on which two dyads you were looking at? And wouldn't the results come out looking just like H.E.?

That's what I'm starting to think now. I just thought the depriming
interval might have been a good way to get started putting this into
regular mapping.
> Maybe you should stick to comparing triads for now, since (IMHO) adding more unique notes to a chord always increases the pain. Do these predictions hold for triads that span intervals other than a 3/2? What about an 8:9:12?

That's what I'm working on now. Note my listening example - the first
two chords are 7:8:9:10:11 and the "minor" version. Sounds pretty
sweet to me.

Geez, the way you talk about this, it's almost like absolutely none of
this has applied to you at all :P Didn't you say the predictions I
made before the listening test accurately described your experience?
Is that not a reason to continue this train of thought? :P

I think I'm just not good enough at communicating new ideas like
this... I tried to overwhelm everyone with all of my barrage of ideas
at once, and it ended up being too long for people to read. So I went
with the listening example, which got more interest, and then I tried
to elucidate my thinking part by part, but the problem now is that it
seems "incomplete" at every step :) I need to find the solution.

> 8:9:12 blows the "lower than 5/4=minor" theory, since I swear on my grandmother's grave that 8:9:12 sounds major to me.

Uh, when did I ever say that? I said that discordance is painful, and
that areas of maximum discordance can be painful, and that painfulness
in small doses is minor. If you flatten the 5/4 to much that it
becomes 8/9, then you aren't in an area of maximum discordance
anymore. I proposed the 60 cents as a rough guideline to find
maximally discordant intervals.

> 9/8 is lower in pitch than 6/5, higher in pain than 7/6, and 8:9:12 has, to my knowledge, lower tonalness than 6:7:9.

4:8:9:12 isn't higher in pain for me than 3:6:7:9.

8:9:12 will have higher tonalness than 6:7:9, since octave
generalization is such a strong habit these days. At least, the
tonalness of 4:8:9:12 is higher than the tonalness of 3:6:7:9. 3:6:7:9
is slightly "bitonal," in that you can hear either 3 as the root, or a
very faint 1.

> I also wanted to believe that, given two triads with the same span, the one with the lower middle note will sound minor in comparison to the one with the higher middle note, but comparing 16:18:21 with its utonal counterpart--1:7/6:21/16--kills that. I'll try to post an audio example tomorrow some time, but to me it's pretty obvious.

That's never what I said. In my last listening example, I gave out
three examples of chords that should have roughly the same amount of
"pain," in a small enough dose that you might term it "sadness," and
only one was minor and the other two had a major third (one had a
supermajor third).

Compare C Eb G to C E F# B D E B, in 12-tet, and tell me if you think
they have similar amounts of sadness. If not, resolve that F# upward
to G, and see if that resolution brings the pain a bit.

-Mike

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Is it as painful if the 5:6:7:9:11:13 part of it slowly decreases in
> volume as the harmonics get higher? I would imagine not.

Less painful, yes, but also the higher harmonics have less of an impact on the feeling.

> I never compared 4:5:6 to 4:5:6:7 or anything like that, nor did I try
> to generalize "majorness."

But isn't majorness the opposite of minorness? How can you define minorness without defining majorness by exclusion? Also, WHY didn't you compare 4:5:6 to 4:5:6:7? It seems a relevant comparison to make.

> I also noticed that sharpening the major third in small doses does the
> same thing, and can create sort of a feeling of sadness too, unless
> it's done too much, at which point it sounds awful, but then starts to
> become harmonic again as the 5/4 becomes 4/3. Hence you have maj9#11
> chords as sounding kind of "sad," and supermaj9#11 chords sounding
> slightly more "sad."

I really don't get a feeling of sadness from sharp major thirds. You may use this datum how you wish.

> > After doing some listening tests, where I used sawtooth, piano, and/or guitar patches, I can conclusively say that 10:13:15 is more discordant to my ears than 14:18:21. I am powerless to explain why this might be, except to hypothesize that it's because the component dyads are simpler in the 14:18:21.
>>
> I really don't hear it that way, but alright. Perhaps there's some
> listener-dependent facet of it. I'd be curious to hear how people
> compare them in a listening test. I can relax and hear 10:13:15 as
> more chilled out after the initial attack. But 14:18:21 bothers me
> more.

Perhaps it's a personality thing?

> I'm not sure how you are abstractly defining "major." I hear 9/7 as
> fitting better into the "Major" sound than 13/10, which is a bit
> further out. But 9/7 just bothers the hell out of me, and 13/10
> doesn't (assuming 3/2 is there as well in both cases).

I suppose I'm defining "major" as "unambiguously happy/bright/cheerful". 10:13:15 has some amount of emotional ambiguity to my ears.

> I don't think that concordance or tonalness leads directly to
> "happiness."

Why not?

> I think that minorness, however, leads to sadness,
> because discordance leads to perceptual pain, and the removal of the
> intrusive stimulus and its replacement by the somewhat cool sound of
> notes merging and fusing together can lead to some sort of "happy"
> feeling after the shrapnel in your ears that is the 13-limit
> "negative" sonority I played in the listening example is removed.

So pleasure is the absence of pain?

> I think that part of the "major" sound that we're used to in 12-tet
> comes from its approximation to 4:5:6, some comes from the major third
> which is slightly sharp and hence a bit "exciting," and part coming
> from how we're used to it being a stable tonic in the sea of
> dissonance and consonance that is the diatonic scale. But I haven't
> really tried to generalize majorness, I'm more on minorness for the
> moment.

I'd hardly call the diatonic scale "a sea of consonance and dissonance". I hear almost no dissonance in the diatonic scale. But I digress.

> I want to reiterate, again, that I no longer think that the 60 cents
> is a hard and fast rule. It was just a nice rule of thumb. See my
> recent letter to Gene for what I really think, if you want to delve
> further into the math. I think it boils down to triadic entropy not
> being dyadic entropy, and I'm working on a transform to calculate it
> for any amount of notes to just put the matter to rest.

Gotcha.

> Is this with triads or bare dyads?

Triads. Always triads.

> > Maybe you should stick to comparing triads for now, since (IMHO) adding more unique notes to a chord always increases the pain. Do these predictions hold for triads that span intervals other than a 3/2? What about an 8:9:12?
> >

> That's what I'm working on now. Note my listening example - the first
> two chords are 7:8:9:10:11 and the "minor" version. Sounds pretty
> sweet to me.

I still think you should figure out triads before piling on the higher harmonics.

> Geez, the way you talk about this, it's almost like absolutely none of
> this has applied to you at all :P Didn't you say the predictions I
> made before the listening test accurately described your experience?
> Is that not a reason to continue this train of thought? :P

Yes, but your example was based entirely on extended-voice/high-harmonic chords where the harmonics were gradually detuned. Saying that "more harmonics=more pain" and "more detuned harmonics=more pain" is pretty obvious, I think. But back in triad land (where I think you should hang out more), things are much less obvious.

> I think I'm just not good enough at communicating new ideas like
> this... I tried to overwhelm everyone with all of my barrage of ideas
> at once, and it ended up being too long for people to read. So I went
> with the listening example, which got more interest, and then I tried
> to elucidate my thinking part by part, but the problem now is that it
> seems "incomplete" at every step :) I need to find the solution.

Well, as a philosophy major, I've got plenty of training in clarifying and streamlining the logic of theoretical ideas. I'd be happy to work with you off-list in making a nice, clean formalization of your hypothesis, once you get it to a point where you're confident in it.

> > 8:9:12 blows the "lower than 5/4=minor" theory, since I swear on my grandmother's grave that 8:9:12 sounds major to me.
>>
> Uh, when did I ever say that? I said that discordance is painful, and
> that areas of maximum discordance can be painful, and that painfulness
> in small doses is minor. If you flatten the 5/4 to much that it
> becomes 8/9, then you aren't in an area of maximum discordance
> anymore. I proposed the 60 cents as a rough guideline to find
> maximally discordant intervals.

Yeah, sorry, you didn't say that, I did...but then I deleted the part where I said it and forgot to clean up the reference. I wrote a whole paragraph of my own hypothesis, then realized it had too many weak-points to be worth advancing at this point and deleted it.

> 4:8:9:12 isn't higher in pain for me than 3:6:7:9.

Where did the 4th and 3rd harmonics come from?

> 8:9:12 will have higher tonalness than 6:7:9, since octave
> generalization is such a strong habit these days. At least, the
> tonalness of 4:8:9:12 is higher than the tonalness of 3:6:7:9.

But I'm not talking about 4:8:9:12 and 3:6:7:9. I'm talking about 8:9:12 and 6:7:9. There's a difference.

> > I also wanted to believe that, given two triads with the same span, the one with the lower middle note will sound minor in comparison to the one with the higher middle note, but comparing 16:18:21 with its utonal counterpart--1:7/6:21/16--kills that. I'll try to post an audio example tomorrow some time, but to me it's pretty obvious.
> >
> That's never what I said. In my last listening example, I gave out
> three examples of chords that should have roughly the same amount of
> "pain," in a small enough dose that you might term it "sadness," and
> only one was minor and the other two had a major third (one had a
> supermajor third).

I know it's never what you said. Okay, here's the incomplete/flawed theory I almost posted, it has a couple of points:
1: to be recognizable as major or minor, a triad must have a span within a certain range, maybe no narrower than 5/4 and no wider than 5/3 (I'm wildly uncertain about this range).
2: a triad will only be recognizable as major or minor in comparison with another triad of equal span (provided both are in the same inversion, and the middle notes are both in the same interval class according to one's internal map).
3: of the pair of triads, the one with the lower middle note will sound minor, and the one with the higher middle note will sound major.

These criteria accurately describe almost all pairs of chords that I've tried, but it breaks down when we look at sus2 chords, which I hear as more major than a minor chord. However, it holds pretty much everywhere else, which is why I'm so confounded.

> Compare C Eb G to C E F# B D E B, in 12-tet, and tell me if you think
> they have similar amounts of sadness. If not, resolve that F# upward
> to G, and see if that resolution brings the pain a bit.

Ugh, I cannot compare these two. It's like apples to screwdrivers.

-Igs