Yahoo Tuning Groups Ultimate Backup tuning Harmonic Entropy Concordance

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13285

> --- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:
> > Hi Paul... and others...
> >
> > I'm not quite getting the tetrad Tenney "ordering" from the
> listening
> > examples at:
> >
> > http://artists.mp3s.com/artists/140/tuning_lab.html
> >
> > It seems that, for example, the tetrad 0___388___702__970, the
> "root
> > position" 4:5:6:7 is really every bit as concordant as you say it
> > should be.
> >
> > However it ranks 22-23! I'm suspicious that something "isn't"
> going
> > on here that should be going on with the audible Tenney
rankings...
>
> > Shouldn't a degree of _audible_ concordance be reflected in
> > these rankings??
>
> Joseph, I told you before -- and you repeated back to me -- that
> these rankings ignore
> otonal/utonal distinctions. As I mentioned in the original tetrad
> post, a true tetradic
> harmonic entropy measure would rank 4:5:6:7 much higher than
> 1/7:1/6:1/5:1/4. But, as
> these two chords contain exactly the same intervals -- diads -- the
> calculation I did
> ranks them in a tie. Sethares would also have to rank them in a tie.
> But clearly there's
> more to it -- John deLaubenfels, take note.

I see, Paul. But my question is more general than that (!!) Why
would this pair be so down the scale at 22-23, when the 4:5:6:7 is so
very concordant it hardly beats at all! At least in "hi fi play"
mode?? The rankings *IN GENERAL* do not sound like increasing
discordance to me (with a few exceptions) ... and I was just
wondering why... (??)
__________ ___ __ __ _
Joseph Pehrson

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13332

>
> Since you're asking about the pair, I assume you're
> including 1/7:1/6:1/5:1/4 in your objection -- that chord
> has the same intervals, and so pretty much the same
> beatings (none?), as 4:5:6:7.

Yes, I am... although I have learned that they are slightly
different. I know, though, that this particular exercise doesn't
differentiate between them...

So, Joseph, which chords in
> the list would you put this 7-limit pair above? Is it just
> the presence of beating that bothers you?

I think so. The 4:5:6:7 sounds so beatless... even weirdly
"electronic" because of the eerie stability of the timbre, that I
would put it almost right at the very top. I also note that you said
this was where that chord was placed in your *own* 22-tEt paper...

Perhaps the
> right ordering for you would be according to the
> _maximum_ discordance of any diad in the chord,
> rather than the _sum_ of the diadic discordances . . . ?
>

Maybe so. I think one particularly discordant diad really *does*
color (literally) my result.

> > At least in "hi fi play"
> > mode?? The rankings *IN GENERAL* do not sound like increasing
> > discordance to me (with a few exceptions) ... and I was just
> > wondering why... (??)
>

> Could you point to some particular departures? There
> are many factors involved.

Once all the files are posted... there are still two more to go up...
I am going to go carefully through and try to group some of them by
"perceived" similarities.... I'm not really sure what I'll come up
with... or even if it will make any sense...

Oh... and I *do* need some text information from you regarding the
two additional tetrads that were *not* originally posted.

I e-mailed you off list about this [SEE, LIST!!!]

Thanks!

_________ ___ __ _
Joseph Pehrson

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13383

>
> Yes, I went along with a "Partchian" philosophy in my
> paper (though allowing tempering), so 4:5:6:7 and 1/7:1/
> 6:1/5:1/4 would be the most concordant tetrads. The
> concepts of my paper were essentially formulated in
> 1991, at age 19. However, I now feel that in many
> respects the 1/7:1/6:1/5:1/4 is clearly not one of the
> more concordant tetrads -- outranked even by other
> half-diminished seventh chords like 5:6:7:9. On the other
> hand, I've grown to like 1/7:1/6:1/5:1/4 for the very
> prominent "guide tone" that appears two octaves
> above the highest note, when using, say, a piano timbre
> -- it's kind of magical, if not exactly what most people
> would call "consonant".

Yes, I note that these particular chords are the ones we had
illustrated in our first "Erlich" tuning example at:

http://artists.mp3s.com/artists/140/tuning_lab.html

I think we pretty much concluded, on the basis of listening, that the
half-diminished otonal 5:6:7:9 really WAS more concordant than the
half-diminished 1/7:1/6:1/5:1/4. It was nice to have this in a
concrete audible form, so some conclusion could be drawn... even if
subjective....

Paul... still don't forget, though, Keenen Pepper is (supposedly) 13!

<Dan's smiley things>
__________ ____ __ _ _
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

>> Quite the opposite -- maybe the above wasn't so
>> obvious after all. For example, the chord 400:512:625 is
>> 5-prime limit, but contains _only_ dissonant intervals
>> (by any reasonable definition). Meanwhile, the
>> saturated chords and otonality and utonality subsets
>> listed above don't have _any_ dissonant intervals within
>> their respective odd limits.

>OK. So then could you please present some kind of summary, if it is
>possible, that would help me understand the idea of "prime" as
>contrasted with "odd" limit as it applies to generalized concordance
>and discordance?? Or does that not pertain??

Hmm . . . in general, prime limit is useless as a measure of concordance.
What matters is something similar to the odd-limit of the individual
intervals, and also otonal/utonal considerations.

However, there are many "crunchy" chords like the major seventh chord which
are pretty consonant and happen to have a low prime limit, since they're
constructed from low odd-limit intervals, and multiplying a bunch of numbers
with a given odd limit will never lead to any higher primes. The major
seventh chord has an odd-limit of 15, but that is only due to one interval,
the 8:15 -- otherwise, it has an odd limit of 5. Clearly the major seventh
chord is more concordant than a chord like 9:11:13:15, which also has an
odd-limit of 15. But chalking up the relative concordance of the major
seventh chord to its low prime limit is clearly a mistake, as chords like
400:512:625 should demonstrate . . .

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13470
>
> Hmm . . . in general, prime limit is useless as a
> measure of concordance. What matters is something
> similar to the odd-limit of the individual intervals,
> and also otonal/utonal considerations.
>
> However, there are many "crunchy" chords like the major
> seventh chord which are pretty consonant and happen to
> have a low prime limit, since they're constructed from
> low odd-limit intervals, and multiplying a bunch of
> numbers with a given odd limit will never lead to any
> higher primes. The major seventh chord has an odd-limit
> of 15, but that is only due to one interval, the 8:15
> -- otherwise, it has an odd limit of 5. Clearly the major
> seventh chord is more concordant than a chord like
> 9:11:13:15, which also has an odd-limit of 15. But
> chalking up the relative concordance of the major
> seventh chord to its low prime limit is clearly a mistake,
> as chords like 400:512:625 should demonstrate . . .

(I've made a webpage of this, for those who can't access the
MIDI-file links from this message)
http://www.egroups.com/files/tuning/monz/prime-odd/prime-odd.htm

I've made MIDI-files of the three chords Paul discusses here:

8:10:12:15, 'classic' 5-prime-limit 15-odd-limit 'major 7th' tetrad
http://www.egroups.com/files/tuning/monz/prime-odd/5l-maj7.mid

9:11:13:15, a 13-prime-limit 15-odd-limit tetrad:
http://www.egroups.com/files/tuning/monz/prime-odd/15o-13p.mid

400:512:625, Paul's highly dissonant 5-prime-limit example triad
http://www.egroups.com/files/tuning/monz/prime-odd/dis5ltri.mid

Here's a triangular lattice diagram of Paul's dissonant
5-limit triad:

25:16
/
/
()
/
/
1:1
/
/
()
/
/
32:25

All tones in this chord lie along the 5-axis. The voicing
1/1 : 32/25 : 25/16 gives the proportions 400 : 512 : 625.
The '8ve'-equivalent prime-factorization of this is
n^0 : 5^-2 : 5^2, or as Graham Breed would write it,
|0 0| |0 -2| |0 2| (the first number is the exponent of 3,
which is always 0 in this case).

I submit that the accordance (relative concordance or
discordance) of a chord is at least partially dependant
on a combination of both the prime- and the exponent-limits
exhibited by the prime-factoring of the ratios of the
perceived intervals in the chord.

(I say 'perceived intervals' to cover the cases where
the tuning is irrational; my opinion is that the ear/brain
system attempts to 'rationalize' musical sounds as much
as it can.)

I'm not arguing that odd-limit has no place in the theory;
but I think more fruitful results would come out of a
combined analysis of the prime-plus-exponent values.

I propose this in my definition of 'sonance', but I
suppose this requires revision, now that we've decided
conclusively that sonance is context-dependant. So
the new entry for 'accordance' should have the contents
of the old 'sonance' definition.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13490
>
> Monz wrote,
>
> >I submit that the accordance (relative concordance or
> >discordance) of a chord is at least partially dependant
> >on a combination of both the prime- and the exponent-limits
> >exhibited by the prime-factoring of the ratios of the
> >perceived intervals in the chord.
>
> And I would submit that an even better measure would totally
> ignore the prime limits and look at the odd-limits of all the
> individual intervals (with Tolerance taken into acccount).

But what about the synergistic otonal component? Wouldn't it
be helpful to model that as a series of harmonics over a
fundamental?, in which case prime-limits gives the *irreducible*
information as to a tone's harmonic 'placement' within the
otonal proportions of the chord.

The fact of the irreducibility of the primes hints to me that
this procedure would rule out the artifacts resulting from
multiple redundant representation of intervals within a given
series.

Looking 'at all the odd-limits of all the individual intervals'
doesn't give any information about the otonal synergy.
...or does it? Did I miss something?

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Monz <MONZ@JUNO.COM>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Monz wrote,

>>>I submit that the accordance (relative concordance or
>>>discordance) of a chord is at least partially dependant
>>>on a combination of both the prime- and the exponent-limits
>>>exhibited by the prime-factoring of the ratios of the
>>>perceived intervals in the chord.

I wrote,

>> And I would submit that an even better measure would totally
>> ignore the prime limits and look at the odd-limits of all the
>> individual intervals (with Tolerance taken into acccount).

Monz wrote,

>But what about the synergistic otonal component?

Yes, that's an additional factor that would make for an still better measure
-- having nothing to do with primality.

>ouldn't it
>be helpful to model that as a series of harmonics over a
>fundamental?

yes.

>in which case prime-limits gives the *irreducible*
>information as to a tone's harmonic 'placement' within the
>otonal proportions of the chord.

Hmm . . . we're back to square one with this . . . the simple fact is that,
unless there are other notes to clue you in, the 23rd harmonic is lower than
the 25th harmonic is lower than the 27th harmonic is lower than the 29th
harmonic, and I have no reason to believe that abstract "reducibility"
affect concordance unless there are additional notes to take advantage of it
by creating some low odd-limit intervals.

>The fact of the irreducibility of the primes hints to me that
>this procedure would rule out the artifacts resulting from
>multiple redundant representation of intervals within a given
>series.

Could you give an example of what you mean?

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/13486

OK... I need a little help in understanding this...

On Monz' new webpage there are three examples...

8:10:12:15, the 'classic' 5-prime-limit 15-odd-limit 'major 7th'
tetrad:

9:11:13:15, a 13-prime-limit 15-odd-limit tetrad

OK, so these are both the SAME ODD limit, but different PRIME limits.

Listening to these two chords, it seems that Monz is right... the
second is clearly more "discordant."

However, then Paul throws a curve with:

400:512:625, Paul's highly dissonant 5-prime-limit example triad

Clearly 5-prime-limit... well, there are no other prime divisors in
there *I* can see...

And yet is is mightily "discordant"

What is going on here??

Does prime limit work or doesn't it???!!!???!?!?!?!?

____________ ___ __ _
Joseph Pehrson

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13504
>
> [me, monz]
> > in which case prime-limits gives the *irreducible*
> > information as to a tone's harmonic 'placement' within the
> > otonal proportions of the chord.
>
> Hmm . . . we're back to square one with this . . . the simple
> fact is that, unless there are other notes to clue you in, the
> 23rd harmonic is lower than the 25th harmonic is lower than
> the 27th harmonic is lower than the 29th harmonic, and I have
> no reason to believe that abstract "reducibility" affect
> concordance unless there are additional notes to take advantage
> of it by creating some low odd-limit intervals.

First, be clear that I'm not talking about dyads here, but
tri- and higher-ads. So I'm not sure why you wrote 'unless
there are other notes to clue you in'.

Well... what I'm thinking about here is that there may be a
'chordal harmonic entropy' where specific *sets* of low-integer
proportions are the 'target', with distribution curves
surrounding both each of the members of a set (as in dyadic
harmonic entropy) *and* the set as a whole (unique to tri-
and higher-ads).

The 'target' aspect of the set comes in when there is a
perceivable numerary nexus, which causes the set to be perceived
as a segment of the harmonic series, giving rise to the
perception of a fundamental (whether present or not).
In order for the numerary nexus to be perceived, the proportions
of the set all need to be quite low-integer.

The prime factors (up to a still-undetermined but quite low
limit, probably around 11, possibly much higher, probably
extremely dependent on context) are the ones that carry the
*unique* perceptual information within a given set of ratios,
*as long as ALL the proportions are low-integer*.

For instance, any low-integer 5-prime-limit ratio can be thought
of as the sum of other low-integer 5-prime-limit ratios, but
11:8 doesn't fit aurally into that paradigm - it doesn't
*sound* like any low-integer 5-limit ratio, or like any
low-integer 7-limit ratio, for that matter. Likewise with
7:4 - it doesn't sound like any low-integer 3-, 5-, or 11-
limit ratio.

I was going somewhere with this, but after doing some experiments,
I lost track of my point, so I'll just post the results here.

I performed some listening tests on two triads, an 8:20:27
and an 8:21:25. Both of them sounded to me like they could
be inversions of minor triads (but I have more to say on this
below): in the first, the 27:8 is an '8ve' of the fundamental,
and in the second, the 21:8 fills that function.

I don't perceive any unique '25-ness' or '27-ness' in either
of them, which says to me that the odd-limit aspect does
not pertain to integers this high in a chordal context.
Partch said as much in his book decades ago.

At the same time, however, I don't perceive any specific
'7-ness' or '5-ness' either. All I hear in both cases is
the relative consonance, but what's really interesting is
that the 8:21:25 *clearly* sounds like a 'minor' chord, while
the 8:20:27 sounds a lot more like an incomplete 'major 6th'
chord (with the '8' giving the fundamental) than an inversion
of a minor chord.

The only things I can ascribe this effect to are the following:

1) The 8:20:27 has the higher odd- and integer-limit, but the
lower prime-limit (= 5). The 8:21:25 has lower odd- and integer-
limit, but higher prime-limit (= 7). So perhaps the 'major-'
and 'minor-ness' inhere in the prime-limit, or perhaps they
inhere in these particular primes, 5 giving 'major' and 7
giving 'minor' (and, had I included it, 11 giving 'neutral'?).
Of course, the 5-limit 10:12:15 certain *sounds* 'minor',
so... uh... ???

2) Since 27 is 3^3, the 'strength' of the 27:8 in reinforcing
the fundamental '8' in the 8:20:27 triad appears to be based on
the 3-prime-limit. On the other hand, perhaps the 'major-ness'
I hear in this chord inheres in the 2:5 between the two lower
notes; the fact that this is 14 cents narrower than the 12-tET
version bothers me not a bit, because of course it's the 'sweet'
5-limit JI 'major 3rd' (plus an '8ve'). The supposedly dissonant
27:20 '4th' (~520 cents) on the top only bothers my sensitive ears
in that it sounds too sharp. I'd bet that this is because of
conditioning, since I'm used to hearing the 500-cent 2^(4/12)
for this interval most of the time.

3) Similarly, in the 8:21:25 triad, the narrowness of the 21:8
'4th' (1 '8ve' + ~471 cents) doesn't bother me, and the 25:21
'minor 3rd' (~302 cents) on top sounds perfectly OK. I suppose,
again, that I'm conditioned to hearing the 300-cent 2^(3/12)
version of this interval, and 25:21 is very close to that.
But why does the 7-limit narrow '4th' sound OK? Hmmm...

Anyway, make of this what you will. Feedback is appreciated.

> [me, monz]
> > The fact of the irreducibility of the primes hints to me that
> > this procedure would rule out the artifacts resulting from
> > multiple redundant representation of intervals within a given
> > series.
>
> [Paul]
> Could you give an example of what you mean?

Hmmm... after thinking about it some more, perhaps I'm really
wrong with this, and what I really mean is exactly the opposite.

Remember my post the other day about the Voronoi diagram? I
mentioned how 3:4:5 is represented redundantly so many times
within the 64-integer-limit. This is because its factors are
the three lowest primes: 2, 3, and 5, which means that more
3-note sets of higher numbers will be multiples of these three
particular notes than of any other set, which results in your
Voronoi diagram having the largest cell for this triad.

I don't know... now I'm really confused. I still think the
prime-factor lattice conveys clear harmonic information about
tri-/higher-ads, but it's a gut feeling, that I'm having
trouble explaining.

The only thing I can say for sure, based on my own experience,
is that discordance of tri-/higher-ads seems to be a function
of the size of both the prime-factors *and* the exponents
which make up the proportions of the chord, and that both
of them need to go into the calculation. (don't ask me *which*
calculation ;-)

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...>
wrote:
> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> http://www.egroups.com/message/tuning/13486
>
> OK... I need a little help in understanding this...
>
> On Monz' new webpage there are three examples...
>
>
> 8:10:12:15, the 'classic' 5-prime-limit 15-odd-limit 'major 7th'
> tetrad:
>
>
> 9:11:13:15, a 13-prime-limit 15-odd-limit tetrad
>
>
> OK, so these are both the SAME ODD limit, but different
> PRIME limits.
>
> Listening to these two chords, it seems that Monz is right... the
> second is clearly more "discordant."
>
> However, then Paul throws a curve with:
>
> 400:512:625, Paul's highly dissonant 5-prime-limit example triad
>
> Clearly 5-prime-limit... well, there are no other prime divisors in
> there *I* can see...
>
> And yet is is mightily "discordant"
>
> What is going on here??
>
> Does prime limit work or doesn't it???!!!???!?!?!?!?
>

Joe, look carefully at the small lattice on that webpage for
this discordant 5-limit triad.

What I've been saying is, according to me, the prime- *AND*
exponent-limits are both important in evaluating accordance
of tri-/higher-ads.

I'll draw another lattice for the first chord.

In the nice, smooth 8:10:12:15 tetrad, '8' is the fundamental,
'10' is the 'major 3rd', '12' is the '5th', and '15' is the
'major 7th'. Here's the lattice:

10 ---- 15
/ \ /
/ \ /
8 ---- 12

10 and 12 are both only one 'step' away from the fundamental 8,
on the 5- and 3-axes respectively. Furthermore, these three
proportions reduce to 4:5:6, the strongest tonal concordance
possible with a tetrad of three distinct identities. 15 is
2 steps away from the fundamental; you can get there by either
of two 'routes': 8 - 10 - 15 or 8 - 12 - 15. In either case,
all relationships are concordant, thus reinforcing the 4:5:6
concordance still further. As Paul pointed out, the only
discordant interval in this chord is the 8:15 between the
'major 7th' and the 'root'.

In the much more discordant 9:11:13:15 tetrad, the perceived
(i.e., reinforced) fundamental is missing from the chord, and
we have 9 (== 3^2) and 15 (= 3*5) both 2 steps away from it,
and 11 and 13 both one step away from it. I would say that
this produces greater discordance.

Something else producing greater discordance in this chord,
already pointed out by both Paul and myself, is the higher
prime-limit. This means that the periodicity between chord-
members is lower: the largest interval, 9:15, reduces to
3:5, which is a dyad with pretty high periodicity. But
the waveforms of the pitches in the dyads involving 11 and
13 have much further to travel before they come back in phase.

But note that while this chord is more discordant than the one
above, it's still not too 'bad' to my ears, and certainly much
more concordant than the following triad. This is probably
because the odd-limit (= 15) is still rather low, and the
integers in the proportion are not much different from the
ones in the first chord.

I'd make a lattice, but it would be difficult following the
triangular formula I used for the 5-limit ones, because of the
number of different prime-axes (i.e., dimensions) involved.
If I had used my own lattice formula, I would have drawn all
three according to that.

Now look again at the lattice of the dissonant triad.
I'll reproduce it here with the proportions instead of
the ratios:

625
/
/
()
/
/
400
/
/
()
/
/
512

It is indeed a 5-limit triad, as can be seen plainly here.
There are several factors which I'd say make it a highly
discordant chord.

First, there are 2 or 4 steps along the 5-axis between all notes
in the chord, with *none* of the notes only one step away from
any other. In this sense, it is much more discordant than
either of the two previous chords.

Second - very closely related to the first point - to use numbers
that give *integer* proportions between all chord-members (which
is what causes the perception of an otonal fundamental) the
numbers have to be quite high, which gives a bad score in the
low-integer game. It's obvious from comparing the proportions
of all three chords that in this sense this one is *far* more
discordant than the other two.

Third, the perceived otonal fundamental (= 512) *is* present in
the chord but is *not* the note functioning as the 'chord root'
(= 400); this creates another kind of 'dissonance' (that between
perceived fundamental and functional chord-root) which I would
say adds further to the discordance.

I might postulate my theory as one based on 'close-packing':
the closer the lattice-points are to the fundamental, the
more concordant the chord. In my lattice formula, higher
primes have longer spaces between steps along their axes
and lower primes have shorter spaces, so 9 (at the second
step on the 3-axis) is still closer to 1/1 than 11 (at the
first step on the 11-axis), and is thus more concordant.

This property does not agree with my perceptions or Paul Erlich's
in 100% of the cases, so my formula still needs fine-tuning
(pun intended); but so far I find it more-or-less workable.

Does that make my theory clearer? And remember, I could be
wrong - these are just my perceptions and speculations.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Monz wrote,

>First, be clear that I'm not talking about dyads here, but
>tri- and higher-ads. So I'm not sure why you wrote 'unless
>there are other notes to clue you in'.

I meant, if there is a 16 and a 25 in the chord, you're pretty much in the
same boat as if you've got a 16 and a 23, _unless_ there is a 20 to create
two nice soft 4:5 consonances.

>Well... what I'm thinking about here is that there may be a
>'chordal harmonic entropy' where specific *sets* of low-integer
>proportions are the 'target', with distribution curves
>surrounding both each of the members of a set (as in dyadic
>harmonic entropy) *and* the set as a whole (unique to tri-
>and higher-ads).

Sounds right.

>The prime factors (up to a still-undetermined but quite low
>limit, probably around 11, possibly much higher, probably
>extremely dependent on context) are the ones that carry the
>*unique* perceptual information within a given set of ratios,
>*as long as ALL the proportions are low-integer*.

Disagree that primes matter for perception.

>I performed some listening tests on two triads, an 8:20:27
>and an 8:21:25. Both of them sounded to me like they could
>be inversions of minor triads (but I have more to say on this
>below): in the first, the 27:8 is an '8ve' of the fundamental,
>and in the second, the 21:8 fills that function.

>I don't perceive any unique '25-ness' or '27-ness' in either
>of them, which says to me that the odd-limit aspect does
>not pertain to integers this high in a chordal context.
>Partch said as much in his book decades ago.

>At the same time, however, I don't perceive any specific
>'7-ness' or '5-ness' either.

Glad you're being honest here!

>All I hear in both cases is
>the relative consonance, but what's really interesting is
>that the 8:21:25 *clearly* sounds like a 'minor' chord, while
>the 8:20:27 sounds a lot more like an incomplete 'major 6th'
>chord (with the '8' giving the fundamental) than an inversion
>of a minor chord.

The only things I can ascribe this effect to are the following:

>1) The 8:20:27 has the higher odd- and integer-limit, but the
>lower prime-limit (= 5). The 8:21:25 has lower odd- and integer-
>limit, but higher prime-limit (= 7). So perhaps the 'major-'
>and 'minor-ness' inhere in the prime-limit, or perhaps they
>inhere in these particular primes, 5 giving 'major' and 7
>giving 'minor' (and, had I included it, 11 giving 'neutral'?).
>Of course, the 5-limit 10:12:15 certain *sounds* 'minor',
>so... uh... ???

>2) Since 27 is 3^3, the 'strength' of the 27:8 in reinforcing
>the fundamental '8' in the 8:20:27 triad appears to be based on
>the 3-prime-limit. On the other hand, perhaps the 'major-ness'
>I hear in this chord inheres in the 2:5 between the two lower
>notes; the fact that this is 14 cents narrower than the 12-tET
>version bothers me not a bit, because of course it's the 'sweet'
>5-limit JI 'major 3rd' (plus an '8ve'). The supposedly dissonant
>27:20 '4th' (~520 cents) on the top only bothers my sensitive ears
>in that it sounds too sharp. I'd bet that this is because of
>conditioning, since I'm used to hearing the 500-cent 2^(4/12)
>for this interval most of the time.

>3) Similarly, in the 8:21:25 triad, the narrowness of the 21:8
>'4th' (1 '8ve' + ~471 cents) doesn't bother me, and the 25:21
>'minor 3rd' (~302 cents) on top sounds perfectly OK. I suppose,
>again, that I'm conditioned to hearing the 300-cent 2^(3/12)
>version of this interval, and 25:21 is very close to that.
>But why does the 7-limit narrow '4th' sound OK? Hmmm...

>Anyway, make of this what you will. Feedback is appreciated.

Nothing in the above suggests any importance in prime limit to triadic
discordance beyond its importance in diadic discordance, nor do you want to
ascribe any importance in prime limit to diadic discordance, do you?

>Hmmm... after thinking about it some more, perhaps I'm really
>wrong with this, and what I really mean is exactly the opposite.

>I don't know... now I'm really confused. I still think the
>prime-factor lattice conveys clear harmonic information about
>tri-/higher-ads, but it's a gut feeling, that I'm having
>trouble explaining.

Yes, the Tenney prime lattice, with 2 included, is the best way to map
pitches according to diadic consonance.

>The only thing I can say for sure, based on my own experience,
>is that discordance of tri-/higher-ads seems to be a function
>of the size of both the prime-factors *and* the exponents
>which make up the proportions of the chord, and that both
>of them need to go into the calculation. (don't ask me *which*
>calculation ;-)

Monz, we can actually come to an agreement here --

*the numbers in each diad, triad, and tetrad can be written as a function of
the prime factors and their exponents in a trivial way -- since you can
factorize any number.

*For simple JI chords, the discordance is a function of the size of the
numbers in each diad, triad, and tetrad.

*For other chords, you need to take tolerance into account.

That's it, there's nothing more to it.

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Monz wrote,

>Something else producing greater discordance in this chord,
>already pointed out by both Paul and myself, is the higher
>prime-limit. This means that the periodicity between chord-
>members is lower: the largest interval, 9:15, reduces to
>3:5, which is a dyad with pretty high periodicity. But
>the waveforms of the pitches in the dyads involving 11 and
>13 have much further to travel before they come back in phase.

Prime-limit is irrelevant to waveform periodicity.

>But note that while this chord is more discordant than the one
>above, it's still not too 'bad' to my ears, and certainly much
>more concordant than the following triad. This is probably
>because the odd-limit (= 15) is still rather low, and the
>integers in the proportion are not much different from the
>ones in the first chord.

I wholly endorse this view.

>Now look again at the lattice of the dissonant triad.
>I'll reproduce it here with the proportions instead of
>the ratios:

> 625
> /
> /
> ()
> /
> /
> 400
> /
> /
> ()
> /
> /
> 512

>It is indeed a 5-limit triad, as can be seen plainly here.
>There are several factors which I'd say make it a highly
>discordant chord.

>First, there are 2 or 4 steps along the 5-axis between all notes
>in the chord, with *none* of the notes only one step away from
>any other. In this sense, it is much more discordant than
>either of the two previous chords.

In other words, it doesn't have any 5-odd-limit intervals, but it has two
25-odd-limit intervals. The 5 doesn't come into the picture at all, it's
only 25 that does! 25 is a high odd number -- and the chord doesn't
approximate a chord with smaller numbers -- so the chord is dissonant.

>Second - very closely related to the first point - to use numbers
>that give *integer* proportions between all chord-members (which
>is what causes the perception of an otonal fundamental) the
>numbers have to be quite high, which gives a bad score in the
>low-integer game. It's obvious from comparing the proportions
>of all three chords that in this sense this one is *far* more
>discordant than the other two.

Yes, the size of the integers is what matters -- if you believe in
octave-equivalence, you get odd-limit directly from that.

But you don't get prime-limit.

>Third, the perceived otonal fundamental (= 512) *is* present in
>the chord but is *not* the note functioning as the 'chord root'
>(= 400); this creates another kind of 'dissonance' (that between
>perceived fundamental and functional chord-root) which I would
>say adds further to the discordance.

Would 512:625:800 be any more consonant? No, the numbers are too high for
the power of two to be perceived as a fundamental.

>I might postulate my theory as one based on 'close-packing':
>the closer the lattice-points are to the fundamental, the
>more concordant the chord.

Even on a Tenney lattice, this would always make an otonal/utonal pair tied
for concordant, which conflict with our perceptions.

>In my lattice formula, higher
>primes have longer spaces between steps along their axes
>and lower primes have shorter spaces, so 9 (at the second
>step on the 3-axis) is still closer to 1/1 than 11 (at the
>first step on the 11-axis), and is thus more concordant.

Tenney does that too, using logs so you get the lengths of composite numbers
like 9 and 15 correctly (9 is not 3+3, but log(9) is log(3) + log(3)).

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13537
>
> Prime-limit is irrelevant to waveform periodicity.

Really?! Can you explain why? I thought periodicity was
dependent on the divisibility of the terms of the proporitions.
Isn't this why Fourier transforms work?

> [about the dissonant 5-limit triad 400:512:625]
> In other words, it doesn't have any 5-odd-limit intervals,
> but it has two 25-odd-limit intervals. The 5 doesn't come
> into the picture at all, it's only 25 that does! 25 is a high
> odd number -- and the chord doesn't approximate a chord with
> smaller numbers -- so the chord is dissonant.

Yes, this is exactly what I was showing with my lattice.
Good way of clarifying my points.

> Yes, the size of the integers is what matters -- if you believe in
> octave-equivalence, you get odd-limit directly from that.
>
> But you don't get prime-limit.

OK - I have to buy that, since I said it before you agreed
with me! :)

> >Third, the perceived otonal fundamental (= 512) *is* present in
> >the chord but is *not* the note functioning as the 'chord root'
> >(= 400); this creates another kind of 'dissonance' (that between
> >perceived fundamental and functional chord-root) which I would
> >say adds further to the discordance.
>
> Would 512:625:800 be any more consonant? No, the numbers are
> too high for the power of two to be perceived as a fundamental.

OK, I can buy that too.

>
> >I might postulate my theory as one based on 'close-packing':
> >the closer the lattice-points are to the fundamental, the
> >more concordant the chord.
>
> Even on a Tenney lattice, this would always make an otonal/utonal
> pair tied for concordant, which conflict with our perceptions.

Hmmm... very good point.

> Tenney does that too, using logs so you get the lengths of
> composite numbers like 9 and 15 correctly (9 is not 3+3,
> but log(9) is log(3) + log(3)).

Makes sense. Like I said, I want to see some Tenney lattices.
Tell me how to make 'em.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Joseph Pehrson <pehrson@pubmedia.com>

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/13520

> At the same time, however, I don't perceive any specific
> '7-ness' or '5-ness' either. All I hear in both cases is
> the relative consonance, but what's really interesting is
> that the 8:21:25 *clearly* sounds like a 'minor' chord, while
> the 8:20:27 sounds a lot more like an incomplete 'major 6th'
> chord (with the '8' giving the fundamental) than an inversion
> of a minor chord.
>
>
> The only things I can ascribe this effect to are the following:
>
> 1) The 8:20:27 has the higher odd- and integer-limit, but the
> lower prime-limit (= 5). The 8:21:25 has lower odd- and integer-
> limit, but higher prime-limit (= 7). So perhaps the 'major-'
> and 'minor-ness' inhere in the prime-limit, or perhaps they
> inhere in these particular primes, 5 giving 'major' and 7
> giving 'minor' (and, had I included it, 11 giving 'neutral'?).
> Of course, the 5-limit 10:12:15 certain *sounds* 'minor',
> so... uh... ???
>

Of course, the "majorness" and "minorness" could just pertain to
these particular examples, yes?? Or is there really a generalized
case for "Prime Limit" determining differences between major and
minor?? Paul???

Yum. How about "Prime Rib"... it's almost lunchtime...

____________ ____ __ _
Joseph Pehrson