Wilson's Constant structure

Is there an agreed upon definition of Wilson's constant structure descriptor? My recollection is that in a constant structure, every instance of a given interval subtends the same number of smaller intervals.

IIRC, there is constant structure scale that is not proper or SP. Can somebody send me the example?

--John

5:6:7 (if I have it right) being a diminished chord and 6:8:9 being a suspended
chord.

Beside the obvious excuse (cultural conditioning)..I get the feeling the fact
5:6:7 has the two prime numbers in it of 5 and 7, despite being much
lower-limit, may have something to do with it.

5:7:9, comes to think of it, has pretty much the same problem (two prime). Do
you believe this...or what else do you think could cause this phenomenon (or do
you, on the other hand, think 5:6:7 IS more consonant than 6:8:9)?

On Tue, Dec 21, 2010 at 12:54 PM, Michael <djtrancendance@...> wrote:
>
> 5:6:7 (if I have it right) being a diminished chord and 6:8:9 being a suspended chord.
>
> Beside the obvious excuse (cultural conditioning)..I get the feeling the fact 5:6:7 has the two prime numbers in it of 5 and 7, despite being much lower-limit, may have something to do with it.
>
> 5:7:9, comes to think of it, has pretty much the same problem (two prime).  Do you believe this...or what else do you think could cause this phenomenon (or do you, on the other hand, think 5:6:7 IS more consonant than 6:8:9)?

5:6:7, in some sense, "fuses" better than 6:8:9. It's easy to hear the
1 pop out when you play 5:6:7, but less easy when you play 6:8:9.

There's also the fact that we often double the root note of these
triads down an octave, which means we're actually comparing 5:10:12:14
and 3:6:8:9. 3:6:8:9 is simpler than 5:10:12:14, and 5:10:12:14 takes
on a much more sinister tone than 5:6:7.

Other than that, I have no idea.

-Mike

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:
>
> Is there an agreed upon definition of Wilson's constant structure
> descriptor?

The definition I put here

http://xenharmonic.wikispaces.com/Periodic+scale

says that interval classes are disjoint. That is, you don't find the same interval in two different classes. If it's wrong, I'd like to know.

On Tue, Dec 21, 2010 at 12:09 PM, John H. Chalmers <JHCHALMERS@...> wrote:
>
> Is there an agreed upon definition of Wilson's constant structure
> descriptor? My recollection is that in a constant structure, every
> instance of a given interval subtends the same number of smaller intervals.
>
> IIRC, there is constant structure scale that is not proper or SP. Can
> somebody send me the example?

Isn't superpyth[7] like that? Every generic interval subtends the same
number of smaller intervals, but the augmented fourth is larger than
the diminished fifth, so it would be improper.

I think it's like
superpyth[7]: improper, CS
meantone[7]: strictly proper, CS
diatonic 7-out-of-12: proper, not CS

With the latter being not CS because the augmented fourth and
diminished fifth are the same size. Am I wrong?

-Mike

On 12/21/2010 12:54 PM, Michael wrote:
> 5:6:7 (if I have it right) being a diminished chord and 6:8:9 being a suspended
> chord.
>
> Beside the obvious excuse (cultural conditioning)..I get the feeling the fact
> 5:6:7 has the two prime numbers in it of 5 and 7, despite being much
> lower-limit, may have something to do with it.
>
> 5:7:9, comes to think of it, has pretty much the same problem (two prime). Do
> you believe this...or what else do you think could cause this phenomenon (or do
> you, on the other hand, think 5:6:7 IS more consonant than 6:8:9)?

With triads you need to consider all intervals. So 5:6:7 has 5:6, 5:7, and 6:7, but 6:8:9 has 3:4, 2:3, and 8:9. Although 8:9 is more dissonant than anything in 5:6:7, the 2:3 and 3:4 are strong consonances. So I don't think it's directly because 5:6:7 has two primes, but since all three are relatively prime, they can't be reduced.

In some ways, though, 5:6:7 seems more "stable" than 6:8:9, which is an aspect of consonance. Certainly it's more stable than other diminished triads, like 25:30:36, which seems to need to resolve to something more consonant. I'd almost want to give it a different name, something like "subdiminished" by analogy to the "subminor" 6:7:9 triad. I think I'd count 6:7:9 as more stable than 5:6:7 though.

Herman>"With triads you need to consider all intervals. So 5:6:7 has 5:6, 5:7,
and 6:7, but 6:8:9 has 3:4, 2:3, and 8:9. Although 8:9 is more dissonant than
anything in 5:6:7, the 2:3 and 3:4 are strong consonances."

Right, so the 6:8:9 has overall higher dyadic consonance (if I get what you
are saying)...which seems to, in many ways, override the higher triadic
"stability" of 5:6:7. The impression I get...is that there are some cases where
dyadic consonance can outweigh triadic consonance...and that an ideal consonance
model would take both into account. And it makes sense: primes in a triad
simply make it less likely the triadic consonance will be high.

On Tue, Dec 21, 2010 at 6:02 PM, Michael <djtrancendance@...> wrote:
>
> Herman>"With triads you need to consider all intervals. So 5:6:7 has 5:6, 5:7, and 6:7, but 6:8:9 has 3:4, 2:3, and 8:9. Although 8:9 is more dissonant than anything in 5:6:7, the 2:3 and 3:4 are strong consonances."
>
>    Right, so the 6:8:9 has overall higher dyadic consonance (if I get what you are saying)...which seems to, in many ways, override the higher triadic "stability" of 5:6:7.  The impression I get...is that there are some cases where dyadic consonance can outweigh triadic consonance...and that an ideal consonance model would take both into account.  And it makes sense: primes in a triad simply make it less likely the triadic consonance will be high.

That seems to be the general consensus. Hopefully I'll get to finish
the entropy-as-integral-transform idea sometime soon, and we'll be
able to see what something like that would look like.

Here's another test for you - what do you hear as more dissonant,
10:13:15 or 14:18:21?

-Mike

I haven't played them but 10:13:15 should be more dissonant than 14:18:21.
The first has 10:13 (dissonant), 10:15 (2/3 consonant) and 13:15 (dissonant). The second has 14:18 (7/9 consonant), 14:21 (2/3 consonant) and 18:21 (6/7 consonant).
So Herman is right.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Dec 21, 2010 at 6:02 PM, Michael <djtrancendance@...> wrote:
> >
> > Herman>"With triads you need to consider all intervals. So 5:6:7 has 5:6, 5:7, and 6:7, but 6:8:9 has 3:4, 2:3, and 8:9. Although 8:9 is more dissonant than anything in 5:6:7, the 2:3 and 3:4 are strong consonances."
> >
> >    Right, so the 6:8:9 has overall higher dyadic consonance (if I get what you are saying)...which seems to, in many ways, override the higher triadic "stability" of 5:6:7.  The impression I get...is that there are some cases where dyadic consonance can outweigh triadic consonance...and that an ideal consonance model would take both into account.  And it makes sense: primes in a triad simply make it less likely the triadic consonance will be high.
>
> That seems to be the general consensus. Hopefully I'll get to finish
> the entropy-as-integral-transform idea sometime soon, and we'll be
> able to see what something like that would look like.
>
> Here's another test for you - what do you hear as more dissonant,
> 10:13:15 or 14:18:21?
>
> -Mike
>

MikeB>"Hopefully I'll get to finish the entropy-as-integral-transform idea
sometime soon, and we'll be
able to see what something like that would look like."

Exactly and....I'm hoping, that model will manage to give triadic "stability"
the advantage in some places and dyadic consonance the advantage in others.

>"Here's another test for you - what do you hear as more dissonant, 10:13:15 or
>14:18:21?"

10:13:15 sounds more dissonant to me...no contest!
How about yourself/selves?
When I actually look at the dyads....

For 14:18:21........
21/14 = 3/2
18/14 = 9/7
21/18 = 7/6
.....some nice simple dyads..........

While for 10:13:15......
15/10 = 3/2
13/10 = 13/10
15/13 = 15/13

Wow...does that second chord ever have high dyadic dissonance (2 of the 3
dyads are 13 odd limit or higher)! Sure, it's more triadic-ally "stable" and
about 3 harmonics lower along the harmonic series...but the huge dyadic
dissonance difference wins over that.

Also, just wondering, do any of you have a good example of a chord where the
dyadic consonance is significantly higher, yet the chord sounds more consonant
due to stability (IE a situation where triadic stability easily trumps dyadic
consonance)?

I'd agree that 10:13:15 is more dissonant, but I will also say that I find 13:15 to be more consonant than 7:9. Frankly, entropy and Tenney-height be damned, I don't hear any dissonance on the whole pitch-spectrum between 10/9 and 6/5...and I'm tempted to say all the way up to about 333 cents, even. Dyadically-speaking, of course. On the other hand, I hear a whole lot of dissonance between 5/4 and 4/3. Anything between about 410 and 466 I hear as quite gnarly. And the dissonance between about 333 and 370 cents is pretty mild in comparison to the dissonance between 7/5 and 3/2.

Of course, all of this is fairly consistent with the predictions of (dyadic) harmonic entropy...I wonder how Steve's calculations compare 10:13:15 to 14:18:21? It's quite interesting how there's sort of a push-pull phenomenon with harmonic-series triads, where those that are really low in the series have low entropy, and those that are much higher in the series can also be low in entropy, but only because they "approximate" (or fall within the field of attraction of) lower-in-the-series triads. 30:38:45, for instance, or 64:81:96, approximate 4:5:6, and are hence lower in entropy than 10:13:15 or 14:18:21. So on the one hand you have Tenney Height (i.e. distance from the root in the harmonic series), and on the other you have absolute pitch-proximity to low-TH intervals.

-Igs

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> I haven't played them but 10:13:15 should be more dissonant than 14:18:21.
> The first has 10:13 (dissonant), 10:15 (2/3 consonant) and 13:15 (dissonant). The second has 14:18 (7/9 consonant), 14:21 (2/3 consonant) and 18:21 (6/7 consonant).
> So Herman is right.
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > On Tue, Dec 21, 2010 at 6:02 PM, Michael <djtrancendance@> wrote:
> > >
> > > Herman>"With triads you need to consider all intervals. So 5:6:7 has 5:6, 5:7, and 6:7, but 6:8:9 has 3:4, 2:3, and 8:9. Although 8:9 is more dissonant than anything in 5:6:7, the 2:3 and 3:4 are strong consonances."
> > >
> > >    Right, so the 6:8:9 has overall higher dyadic consonance (if I get what you are saying)...which seems to, in many ways, override the higher triadic "stability" of 5:6:7.  The impression I get...is that there are some cases where dyadic consonance can outweigh triadic consonance...and that an ideal consonance model would take both into account.  And it makes sense: primes in a triad simply make it less likely the triadic consonance will be high.
> >
> > That seems to be the general consensus. Hopefully I'll get to finish
> > the entropy-as-integral-transform idea sometime soon, and we'll be
> > able to see what something like that would look like.
> >
> > Here's another test for you - what do you hear as more dissonant,
> > 10:13:15 or 14:18:21?
> >
> > -Mike
> >
>

Igs>"It's quite interesting how there's sort of a push-pull phenomenon with
harmonic-series triads, where those that are really low in the series have low
entropy, and those that are much higher in the series can also be low in
entropy, but only because they "approximate" (or fall within the field of
attraction of) lower-in-the-series triads. 30:38:45, for instance, or
64:81:96, approximate 4:5:6"

If that were true, in this case, what does 14:18:21 approximate?
Off the top of my head 5:6:7 and 7:9:11 seem the obvious candidates...and we've
already agreed 5:6:7 sounds less consonant.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Igs>"It's quite interesting how there's sort of a push-pull phenomenon with
> harmonic-series triads, where those that are really low in the series have low
> entropy, and those that are much higher in the series can also be low in
> entropy, but only because they "approximate" (or fall within the field of
> attraction of) lower-in-the-series triads. 30:38:45, for instance, or
> 64:81:96, approximate 4:5:6"
>
> If that were true, in this case, what does 14:18:21 approximate?
> Off the top of my head 5:6:7 and 7:9:11 seem the obvious candidates...and we've
> already agreed 5:6:7 sounds less consonant.

Let me stress that I use the word "approximate" here in the most literal sense, somewhat divorced from the connotations typically associated with it here. I use it to mean "to be within the proximity of", in reference only to the pitch-distance between two triads. I don't attach the connotations of "bears psychoacoustic resemblance to", "can substitute for", or "gives the impression of". 14:18:21 is in proximity to 4:5:6, more so than 10:13:15, and it's a known fact that being in proximity to a low-entropy triad is associated with a decrease in entropy. It's also a known fact that being equidistant from two low-entropy triads is usually associated with being high entropy, and 10:13:15 is almost half-way between 4:5:6 and 6:8:9.

At any rate, it may be fair to say that 14:18:21 approximates (i.e. is in the proximity of) 7:9:11 (or possibly 5:6:7) as well, but I think that it is its proximity to 4:5:6 that gives it the biggest drop in entropy. I say this mostly because we describe it qualitatively as a "supermajor" triad, suggesting that we interpret it as a modified major triad. Of course, I will admit that musicians for whom I have played supermajor triads have said both "it sounds like an augmented chord, sort of" and "it sounds almost like a diminished chord", though when alternated with a regular major triad most hear it as a sort of "half-suspension" and when alternated or used in progressions with subminor triads, it just sounds "like a nasty major chord". So my intuition says that the proximity to all three of those triads (4:5:6, 5:6:7, 7:9:11) can be felt, depending on the listener's mental state and the degree of priming. I'm fairly certain the timbre will be important as well.

-Igs

Igs>"At any rate, it may be fair to say that 14:18:21 approximates (i.e. is in
the proximity of) 7:9:11 (or possibly 5:6:7) as well, but I think that it is
its proximity to 4:5:6 that gives it the biggest drop in entropy."

I think I follow what you are saying, but it comes across as odd to me as
14:18:21 sounds nothing like 4:5:6 to my ears and, as well, 5:4 and 9:7 also
have completely distinct characters to my ears as individual dyads.

So while you seem to be making a point of saying 9/7 is near 5/4 far as pitch
distance...I don't see the gravitation. To me 9/7 is its own center of sorts,
and not a "weak/estimated 5/4" or anything like that. Same goes with 7/6 vs.
6/5 between the two chords...to me 7/6 is it's own identity and not a "weak"
6/5. You seem to be stating an opinion that Harmonic Entropy (and
mostly/"only?" the low-entropy areas it favors) is the answer...and I have my
doubts.

>"my intuition says that the proximity to all three of those triads (4:5:6,
>5:6:7, 7:9:11) can be felt, depending on the listener's mental state and the
>degree of priming."
That is believable it can happen, but I doubt very often IE when is it "easy"
to simply swap a major chord with a diminished one without changing the mood
significantly?

And well...let's say your "pitch proximity" idea does hold in most cases...how
could that be used to help a musician pick a chord for a desired emotion (rather
than using, say, the "can substitute for" definition myself and others often use
for chord likeness)? Maybe if you can find patterns in how priming "makes
chords act/be heard like other chords" it could help...I, for one though, can't
think of any easy examples of such priming.

On Wed, Dec 22, 2010 at 12:30 AM, cityoftheasleep
<igliashon@...> wrote:
>
> I'd agree that 10:13:15 is more dissonant, but I will also say that I find 13:15 to be more consonant than 7:9

Do you mean you hear 10:13 to be more consonant than 7:9? In any case,
I'm the complete opposite of you guys - I hear 10:13:15 as more
consonant and 7:9 more consonant than 10:13.

I guess 7:9 and 10:13 sound consonant in different ways: 7:9 is more
biting, but fuses better, and 10:13 is less biting, but doesn't fuse
as well.

-Mike

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> I think I follow what you are saying, but it comes across as odd to me as
> 14:18:21 sounds nothing like 4:5:6 to my ears and, as well, 5:4 and 9:7 also
> have completely distinct characters to my ears as individual dyads.
>
> So while you seem to be making a point of saying 9/7 is near 5/4 far as pitch
> distance...I don't see the gravitation. To me 9/7 is its own center of sorts,
> and not a "weak/estimated 5/4" or anything like that. Same goes with 7/6 vs.
> 6/5 between the two chords...to me 7/6 is it's own identity and not a "weak"
> 6/5. You seem to be stating an opinion that Harmonic Entropy (and
> mostly/"only?" the low-entropy areas it favors) is the answer...and I have my
> doubts.

Aaargh...Michael, did you read ANYTHING of what I said of how I use the term "approximates"? I specifically stated that I use the term *only* to refer to pitch-distance, and that it does not mean "sounds like", "can be substituted for", "is psychoacoustically similar to", etc. etc.

I do not believe ratios have anything to do with identities. Some ratios are audibly beatless, others are not, but beatlessness is not the defining principle of intervallic identity. Saying that 9/7 is a "type of major 3rd" is in no way saying it's a "bad/weak 5/4". If you look at the range of what can function as a major 3rd, you'll notice that 5/4 is near-about in the center, which can be taken to mean whatever you like.

> >"my intuition says that the proximity to all three of those triads (4:5:6,
> >5:6:7, 7:9:11) can be felt, depending on the listener's mental state and the
> >degree of priming."
> That is believable it can happen, but I doubt very often IE when is it "easy"
> to simply swap a major chord with a diminished one without changing the mood
> significantly?

Well, the point is that the supermajor triad bears some similarity to multiple triads at once, so it's not as if anything is being "swapped"...more like different aspects can be noticed under different circumstances.

> And well...let's say your "pitch proximity" idea does hold in most cases...how
> could that be used to help a musician pick a chord for a desired emotion (rather
> than using, say, the "can substitute for" definition myself and others often use
> for chord likeness)? Maybe if you can find patterns in how priming "makes
> chords act/be heard like other chords" it could help...I, for one though, can't
> think of any easy examples of such priming.

Well, a musician never picks just one chord for a desired emotion. But I will say this: start with a triad of root-4th-5th (i.e. like a 6:8:9), then drop the 4th down to about 450 cents. The 450-cent interval will sound somewhat like a major 3rd. Clear your ears for a minute, then play a major triad. Then move the major 3rd *up* to about 450 cents. The 450-cent interval will sound somewhat more like a 4th.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Do you mean you hear 10:13 to be more consonant than 7:9?

No, I meant what I said. I find 10:13 less consonant than 7:9 and 13:15, but 13:15 more consonant than both. I said this because 13:15 is higher in TH than 7:9. I find quite a lot of intervals more consonant than 7:9, including 8:9, 7:11, and 11:13.

> In any case,
> I'm the complete opposite of you guys - I hear 10:13:15 as more
> consonant and 7:9 more consonant than 10:13.
> I guess 7:9 and 10:13 sound consonant in different ways: 7:9 is more
> biting, but fuses better, and 10:13 is less biting, but doesn't fuse
> as well.

I wouldn't call either of them consonant.

-Igs

On Wed, Dec 22, 2010 at 5:51 PM, cityoftheasleep
<igliashon@...> wrote:
>
> No, I meant what I said. I find 10:13 less consonant than 7:9 and 13:15, but 13:15 more consonant than both. I said this because 13:15 is higher in TH than 7:9. I find quite a lot of intervals more consonant than 7:9, including 8:9, 7:11, and 11:13.

OK, I see.

> > In any case,
> > I'm the complete opposite of you guys - I hear 10:13:15 as more
> > consonant and 7:9 more consonant than 10:13.
> > I guess 7:9 and 10:13 sound consonant in different ways: 7:9 is more
> > biting, but fuses better, and 10:13 is less biting, but doesn't fuse
> > as well.
>
> I wouldn't call either of them consonant.

I hear them all as pretty consonant, but I guess we're back to the
usual debate on what factors of the sound you're hearing you define as
"consonant." I don't hear 9/7 as a hopelessly sharp 5/4 or anything
like that, and I have recently fallen in love with 13/10. Of the two,
when a fifth is added, I prefer 10:13:15 to 14:18:21 - 14:18:21 I hear
more as a "major triad with a hopelessly sharp 5/4" than 10:13:15,
where it sounds more like the major third is so sharp that it finally
hits a "new" interval instead of getting sucked in constantly to the
5/4.

-Mike

Herman wrote:

> With triads you need to consider all intervals. So 5:6:7 has
> 5:6, 5:7, and 6:7, but 6:8:9 has 3:4, 2:3, and 8:9. Although
> 8:9 is more dissonant than anything in 5:6:7, the 2:3 and
> 3:4 are strong consonances. So I don't think it's directly
> because 5:6:7 has two primes, but since all three are
> relatively prime, they can't be reduced.
>
> In some ways, though, 5:6:7 seems more "stable" than 6:8:9,
> which is an aspect of consonance.

I agree completely and would just like to add that I am
flirting more and more with the notion that dyadic composition
ceases to be important in the case of sine tones. Of course
we almost never hear clean sine tones. But here's my shot
at synthesizing them:

http://lumma.org/temp/567.wav
http://lumma.org/temp/689.wav

-Carl

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

> http://lumma.org/temp/567.wav
> http://lumma.org/temp/689.wav

I hope I don't push over anyone's apple cart, but 567 sounds distinctly more consonant to me.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I agree completely and would just like to add that I am
> flirting more and more with the notion that dyadic composition
> ceases to be important in the case of sine tones. Of course
> we almost never hear clean sine tones. But here's my shot
> at synthesizing them:
>
> http://lumma.org/temp/567.wav
> http://lumma.org/temp/689.wav

5:6:7 for the win here.

I think that what you're saying about dyadic resonances not mattering with sine waves is true, but that this property naturally emerges out of another property, which is that the entirety of everything you're hearing, at all, is a bunch of sine waves added together, or at least as far as the cochlea is concerned.

Keep in mind that virtual fundamental placement is occurring not just for the triad you're listening to, but for the timbres of all of the notes in the triad as well. If you play 4:5:6 with timbres in which 1 is removed, the harmonic series first refer to those 1's, and then the fundamental 1's then recursively point to a double virtual fundamental 1 for the 4:5:6 triad.

What seems to be somewhat important is that the multiple fundamentals themselves are harmonically related (which turns out to be the same thing as prioritizing "triadic consonance" over "dyadic consonance").

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I hear them all as pretty consonant, but I guess we're back to the
> usual debate on what factors of the sound you're hearing you define as
> "consonant." I don't hear 9/7 as a hopelessly sharp 5/4 or anything
> like that, and I have recently fallen in love with 13/10. Of the two,
> when a fifth is added, I prefer 10:13:15 to 14:18:21 - 14:18:21 I hear
> more as a "major triad with a hopelessly sharp 5/4" than 10:13:15,
> where it sounds more like the major third is so sharp that it finally
> hits a "new" interval instead of getting sucked in constantly to the
> 5/4.

Yes, right around 13/10 seems to be where the gravity-well of the major-3rd category is escaped, and I'd give it about a 5-cent margin on either side (so about 450 to 460 cents) where it hovers in "neither 3rd nor 4th" territory. However, I do object to saying the 9/7 in a 14:18:21 triad is a "hopelessly sharp 5/4", because I haven't heard a good argument for why anything heard as a major 3rd in a triad must be some (per)version of a 5/4.

-Igs

On Wed, Dec 22, 2010 at 9:05 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Yes, right around 13/10 seems to be where the gravity-well of the major-3rd category is escaped, and I'd give it about a 5-cent margin on either side (so about 450 to 460 cents) where it hovers in "neither 3rd nor 4th" territory. However, I do object to saying the 9/7 in a 14:18:21 triad is a "hopelessly sharp 5/4", because I haven't heard a good argument for why anything heard as a major 3rd in a triad must be some (per)version of a 5/4.
>
> -Igs

If I construct a timbre out of sine waves in a 14:18:21 frequency, and
play happy birthday to you, the fundamental that pops out is the same
as if it were a really mistuned 4:5:6.

Nonetheless, we've been over this all before: I have no idea what the
big picture is and I don't think that anyone really does. 14:18:21
sounds like the major third is uncomfortably sharp, and 10:13:15
sounds like the major third has gotten so sharp that it sounds like
some other new thing. I don't think that the 7:9 in 7:9:11 will sound
like a mistuned 4:5, but I think that for 14:18:21 it just may.

-Mike

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
> Of course, all of this is fairly consistent with the predictions of (dyadic) harmonic entropy...I wonder how Steve's calculations compare 10:13:15 to 14:18:21?

10:13:15 or (454,248) in lower,upper cents has THE of 5.8616.
It is within 3 cents of a local minimum at (451,250) with THE of 5.6811.
44 cents from a local minimum at approx 6:8:9

14:18:21 or (435,267) has THE of 5.8694.
It is 16 cents from the above local minimum,
44 cents from a local minimum at approx 11:14:16
49 cents from 4:5:6

By the way, the median of the calculated THE values on the grid is 5.8781.

Steve M.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I think it's like
> superpyth[7]: improper, CS
> meantone[7]: strictly proper, CS
> diatonic 7-out-of-12: proper, not CS
> With the latter being not CS because the augmented fourth and
> diminished fifth are the same size. Am I wrong?

I think that's right; that is assuming "7-out-of-12" implies 12EDO. In fact it seems to me that with a generator of exactly 700 cents, the 7-note MOS scale is not a CS, but with a generator slightly larger or smaller, it is a CS (no need to go to Pyth, let alone superpyth). Am I wrong?

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> > Of course, all of this is fairly consistent with the predictions of (dyadic) harmonic entropy...I wonder how Steve's calculations compare 10:13:15 to 14:18:21?
>
> 10:13:15 or (454,248) in lower,upper cents has THE of 5.8616.
> It is within 3 cents of a local minimum at (451,250) with THE of 5.6811.
> 44 cents from a local minimum at approx 6:8:9

Why here? How many such local minima are here?

Igs>"Aaargh...Michael, did you read ANYTHING of what I said of how I use the
term "approximates"? I specifically stated that I use the term *only* to
refer to pitch-distance, and that it does not mean "sounds like", "can be
substituted for", "is psychoacoustically similar to", etc. etc."

Or did you even read my answer?! My reply was
Me>"So while you seem to be making a point of saying 9/7 is near 5/4 far as
PITCH DISTANCE...I don't see the gravitation."

IE I UNDERSTAND you ARE talking about pitch distance when you say
"approximates"...I just don't see how pitch distance is relevant to how
musicians can work with/use odd chords.

You also went on do describe how you think how a chord is perceived or what it
"sounds like" is based much on priming rather than the chord itself...but never
explained why.

>"Saying that 9/7 is a "type of major 3rd" is in no way saying it's a "bad/weak
>5/4"."
But your argument still seems to be....we hear 5/4 and 9/7 of different
variations of the same compositional "idea" of a major third. If you are also
NOT saying 9/7 is a bad/weak 5/4 or, rather, a "weaker" major third...than what
are you saying, what's the musical implication?

>If you look at the range of what can function as a major 3rd, you'll notice
>that 5/4 is near-about in the center"
Again, you seem to be saying 9/7 is at the "edge of a range"...indirectly
implying it's weaker. Confusing....

>"Well, the point is that the supermajor triad bears some similarity to multiple
>triads at once, so it's not as if anything is being "swapped"...more like
>different aspects can be noticed under different circumstances. "
Ok, now this is starting to make sense.
So, the further away IN PITCH DISTANCE from a "tonal center" like 5/4 more
dyads in a chord are...the more likely a chord can swap musical purposes (not to
say the chord is necessarily weaker but, instead, more "swappable")? That I
certainly agree with...

The part I don't get is a chord fairly far, even in pure pitch distance, from
a 4:5:6 like a 14:18:21's gravitating toward sounding like the 4:5:6 a huge
majority of the time (if that's what you are saying), even with priming. Also,
who says it won't swap with, say, the 10:13:15 chord instead (since that chord
IS closer in pitch distance than 4:5:6 is)? Unless you ARE again assuming
tonal gravitation toward 5/4...which is again why I'm thinking you are pushing
things based on Dyadic harmonic entropy rather than the "Pitch Distance" idea
you claim to be championing.

I believe it usually sounds like it's own identity by itself but, as you said,
can be "primed" to act as a 4:5:6...but not very easily so. In fact, my
Untwelve composition uses the 18:22:27 chord a lot IE 1/1 11/9 3/2....and the
emotional feel rarely rounds to either 4:5:6 or 10:12:15...maybe around 1/4 of
the time.

In short, I think you are overestimating the ability of "unstable" chords to
be swappable with stable ones, even with priming involved....and am thinking if
you really were talking about pitch distance rather than "gravitation toward
more stable chords"...you would say 14:18:21 approximates (in your terminology)
10:13:15...because the pitch distance between those two is lower than 14:18:21
and 4:5:6.

>"But I will say this: start with a triad of root-4th-5th (i.e. like a 6:8:9),
>then drop the 4th down to about 450 cents. The 450-cent interval will sound
>somewhat like a major 3rd. Clear your ears for a minute, then play a major
>triad. Then move the major 3rd *up* to about 450 cents. The 450-cent interval
>will sound somewhat more like a 4th."
I hear you...will have to try this when I get home...

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

> Why here? How many such local minima are here?
>

Should read "How many such local minima are there?"

On Thu, Dec 23, 2010 at 5:12 AM, martinsj013 <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > I think it's like
> > superpyth[7]: improper, CS
> > meantone[7]: strictly proper, CS
> > diatonic 7-out-of-12: proper, not CS
> > With the latter being not CS because the augmented fourth and
> > diminished fifth are the same size. Am I wrong?
>
> I think that's right; that is assuming "7-out-of-12" implies 12EDO. In fact it seems to me that with a generator of exactly 700 cents, the 7-note MOS scale is not a CS, but with a generator slightly larger or smaller, it is a CS (no need to go to Pyth, let alone superpyth). Am I wrong?

Right, out of 12-edo. So an improper scale and a strictly proper scale
can be CS, but a regular proper scale can't.

-Mike

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

> >"Saying that 9/7 is a "type of major 3rd" is in no way saying it's a "bad/weak
> >5/4"."
> But your argument still seems to be....we hear 5/4 and 9/7 of different
> variations of the same compositional "idea" of a major third. If you are also
> NOT saying 9/7 is a bad/weak 5/4 or, rather, a "weaker" major third...than what
> are you saying, what's the musical implication?

There is no musical implication. This conversation was about harmonic entropy, which has nothing to do with music. I hypothesized that 14:18:21 would have lower entropy than 10:13:15 on account of the fact that 14:18:21 approximates 4:5:6 better, i.e. is closer to it and possibly subject to a drop in entropy due to being in 4:5:6's entropy-well. Steve's calculations, however, have shown this not to be the case. 10:13:15 and 14:18:21 are pretty close in entropy.

> >If you look at the range of what can function as a major 3rd, you'll notice
> >that 5/4 is near-about in the center"
> Again, you seem to be saying 9/7 is at the "edge of a range"...indirectly
> implying it's weaker. Confusing....

Indirectly implying NOTHING. You are confused because you keep superimposing your own ideas and prejudices on what I type, despite the fact that I disagree with them.

I do not think of intervals in terms of "strong" and "weak", I think of them in terms of mood and quality. I reject intervallic hierarchies, ideas such as "error" and "mistuning". There are no bad intervals.

> >"Well, the point is that the supermajor triad bears some similarity to multiple
> >triads at once, so it's not as if anything is being "swapped"...more like
> >different aspects can be noticed under different circumstances. "
> Ok, now this is starting to make sense.
> So, the further away IN PITCH DISTANCE from a "tonal center" like 5/4 more
> dyads in a chord are...the more likely a chord can swap musical purposes (not to
> say the chord is necessarily weaker but, instead, more "swappable")? That I
> certainly agree with...

Yes, now you have got it. We are acculturated to categorize intervals in a certain way, and to categorize chords in certain ways. While we microtonalists tend to have more categories at our disposal into which we can place intervals and chords, naive common-practice-trained musicians (and most average listeners) don't. So microtonal chords--like the supermajor triad--which are somewhat between "normal" categories will cause confusion to many listeners if they hear the chord "nakedly". Given some context, however, they'll be able to "figure it out". A good analogy to this phenomenon is to take a familiar word and misspell it horrendously, even to the point of just jumbling up the letters. Viewed on its own, the word will look nonsensical. But put it in a sentence where the properly-spelled word would make sense, and the mind is able to "correct it". Heck, you can take an entire paragraph and jumble the letters in each word, but it will still ultimately be legible because the relationship of each jumbled word to its neighbors gives enough cue for the mind to "puzzle it out".

Now, to relate this back to "swappability": ever here of synanagrams? They're different words made up of the same letters, like leaf and flea or players/replays/parsley. So imagine you take a word that has one or two syanagrams of it, and jumble up the letters. Unless you put the jumbled letters in a sentence where only one of the synangrams will make sense, it will be difficult to un-jumble the letters. So, effectively, that jumble of letters will be multiple words at once, and can switch between them depending on the reader's interpretation. This, essentially, is what chords like neutral or supermajor triads are to some naive/average musicians/listeners. To non-naive (i.e. microtonally-aware) listeners/musicians, this is not the case--the "jumble of letters" is a word in itself, i.e. it carries meaning which can be deciphered independently of context.

> The part I don't get is a chord fairly far, even in pure pitch distance, from
> a 4:5:6 like a 14:18:21's gravitating toward sounding like the 4:5:6 a huge
> majority of the time (if that's what you are saying), even with priming. Also,
> who says it won't swap with, say, the 10:13:15 chord instead (since that chord
> IS closer in pitch distance than 4:5:6 is)? Unless you ARE again assuming
> tonal gravitation toward 5/4...which is again why I'm thinking you are pushing
> things based on Dyadic harmonic entropy rather than the "Pitch Distance" idea
> you claim to be championing.

Okay, look. In my way of thinking, ratios do not define interval classes. Saying that one ratio is "heard as" another is nonsense. Ratios are a way of designating pitches--they are discrete, specific, exact, not wide enough to define a perceptual class. It's as absurd to me to say that 14:18:21 is "heard as" 4:5:6 as it is to say that "the number 2 is computed as the number 4". Just because two ratios sound similar to each other, does not mean that there is any basis for saying that one ratio is "heard as" the other. They are *both* heard as members of a larger and more general class. Classes are fuzzy, they overlap each other, but they also have "centers" of clarity--which are determined by the listener's musical acculturation. The number of classes and the fuzziness/overlap of their boundaries is entirely listener-dependent. As I said above, 5/4 is "near" the center of what most listeners recognize as a major 3rd, but it is not *in* the center. That would be 400 cents. I say this because most musicians I've played a pure 5/4 for have said it sounds "a little flat".

For the purposes of categorizing 14:18:21 and 10:13:15, I am operating using standard diatonic chord classes. Both of these chords are on the continuum between "major triad" and "suspended 4th triad". 14:18:21 is significantly closer to the major triad than it is to the suspended 4th triad, whereas 10:13:15 is just about smack in the middle between them and could be interpreted as both or neither.

On the other hand, if we allow for more detailed intervallic categorization, and assume a listener that is as acculturated to something like 24-EDO as we are to 12-EDO, then there will be another chord-class between the major triad and the suspended 4th triad, though I'm at a loss to name it. For this listener accustomed to hearing a 0-450-700-cent triad, 14:18:21 will without a doubt be absorbed into *that* chord-class, rather than the class of "major triad".

For me, with my combination of western acculturation and microtonal sensitivity, I consider "supermajor triad" to be a distinct class, and assuming a constant-size of 5th, the range of 3rds I consider "supermajor" runs from about 420 to 450 cents, which has 9/7 somewhere in the center. Above 450 but below about 475, I'd say there's another class of "suspended flat-4th", which has 21/16 somewhere near the center. So despite my microtonal sensitivity, I still find quarter-tone(ish) intervals right on the boundary between interval classes. YMMV.

-Igs

Igs>"There is no musical implication. This conversation was about harmonic
entropy, which has nothing to do with music. I hypothesized that 14:18:21
would have lower entropy than 10:13:15"
Why oh why does every discussion I start on consonance have to be
pigeon-holed into Harmonic Entropy (especially when I don't mention HE)? For
the record, not everyone agrees Harmonic Entropy has any sort of monopoly on
defining consonance...

>"You are confused because you keep superimposing your own ideas and prejudices
>on what I type, despite the fact that I disagree with them."
I'm asking about consonance with regard to degree of relaxation between
chords, not limited to HE. :-S You are changing the topic...no wonder your
consensus disagrees with mine (as it's on a difference topic!). If you were on
topic...you should be able to explain why you don't think 9/7 is "on the edge of
a major 3rd"...it seems you are simply dodging the question...

>"I do not think of intervals in terms of "strong" and "weak", I think of them in
>terms of mood and quality."
Again, I'm talking in terms of degree of relaxation between the chords, not
"mood". I thought that would be obvious when I used the term consonance and not
entropy...but I guess not. Again, you are changing the question...I'm here to
solve the very question you seem to be set on avoiding, that of musical
implication IE which chords can be used better as points of resolve/relaxation.

Me>> So, the further away IN PITCH DISTANCE from a "tonal center" like 5/4
more

>> dyads in a chord are...the more likely a chord can swap musical purposes (not
>>to
>>
>> say the chord is necessarily weaker but, instead, more "swappable")? That I

>> certainly agree with...
Igs>Yes, now you have got it.
That works... :-) It brings us back to the topic of musical
implication...even if we are talking more about usage than degree of
relaxation. It's also the one sub-topic of Harmonic Entropy that seems to
relate to the main topic I introduced.

...Will continue to answer this (and try your example) later..........