approximation?

I was listening some different minor tetrads (the 7-limit utonality, the
two ASS candidates, and one or two others), and paying close attention to
where the fundamental seemed to be. When I seemed to catch the 1/1 6/5 3/2
7/4 chord approximating a 5-6-15-9 chord on 8/5!

Besides a vague sound at 8/5, the 7/4 seems to get sharper as I go from 1/1
5/4 3/2 7/4 to 1/1 6/5 3/2 7/4! Thinking maybe the dissonant 35/24 was
throwing things off, I compared the 7/4 - 5/4 and the 7/4 - 6/5 dyads. And
noticed no change in the 7/4.

This is the most glaring example of an Erlich-type approximation I have
ever heard. Can anyone confirm or deny?

Carl

> I was listening some different minor tetrads
> (the 7-limit utonality, the two ASS candidates,
> and one or two others), and paying close attention to
> where the fundamental seemed to be. When I
> seemed to catch the 1/1 6/5 3/2 7/4 chord
> approximating a 5-6-15-9 chord on 8/5!
>
> Besides a vague sound at 8/5, the 7/4 seems to get
> sharper as I go from 1/1 5/4 3/2 7/4 to 1/1 6/5 3/2 7/4!
> Thinking maybe the dissonant 35/24 was throwing things
> off, I compared the 7/4 - 5/4 and the 7/4 - 6/5 dyads.
> And noticed no change in the 7/4.
>
> This is the most glaring example of an Erlich-type
> approximation I have ever heard. Can anyone confirm
> or deny?

This is neither confirmation nor denial, but observation.
It makes sense to me that the 7/4 would seem to
go sharp and approximate the 9/5. This seems to be
a confirmation of the "tonalness" idea.

Both chords are implying the same fundamental, 8/5.
But the one with 9/5 has proportions (for the total
chord) that are *half* those of the chord with 7/4:

[8/5] 1/1 5/4 3/2 9/5
[8:] 10 : 12 : 15 : 18
= [16:] 20 : 24 : 30 : 36

[8/5] 1/1 5/4 3/2 7/4
[16:] 20 : 24 : 30 : 35

I think the fact that the 9/5 makes a chord with such
smaller proportions, causes a tendency for us (or at
least for you in this experiment) to perceive
the "tonalness" in the 7/4 chord as the simpler model
of 1/1 5/4 3/2 9/5.

Added weight is given to this explanation by the
fact that you tested dyads separately and did not
find evidence of this perceptual sharpening there.

It would come about only as a result of the stronger
implication of a fundamental, in chords of 3 or more
tones. Dyads are much more ambiguous in their
tonal implications.

- Monzo
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Joseph L Monzo wrote:

> [8/5] 1/1 5/4 3/2 9/5
> [8:] 10 : 12 : 15 : 18
> = [16:] 20 : 24 : 30 : 36
>
> [8/5] 1/1 5/4 3/2 7/4
> [16:] 20 : 24 : 30 : 35

Excuse me, is'nt it...

[8/5] 1/1 5/4 3/2 9/5
[16:] 20 : 25 : 30 : 36

[8/5] 1/1 5/4 3/2 7/4
[16:] 20 : 25 : 30 : 35

David Mezquita.

Carl,

your post would have been easier to follow (for me at least) if you had
used some convention like: fractions x/y for names of single notes, and
ratios x:y, x:y:z for dyads, triads etc. Of course in the case of triads
and higher, one can have ratios between fractions like a/b : c/d : e/f. I
know I've been guilty of the same sloppy notation at times.

I also think we should stop writing dyad ratios in the reverse order to how
we write higher-ads (not that you did Carl). While the note 5/4 is quite
diferent to 4/5. The interval 4:5 is just the same as 5:4 and it seems
weird to write that 4:5:6 contains 5:4. So...

Carl Lumma <clumma@nni.com> wrote:
>I was listening some different minor tetrads (the 7-limit utonality, the
>two ASS candidates, and one or two others), and paying close attention to
>where the fundamental seemed to be. When I seemed to catch the 1/1 6/5 3/2
>7/4 chord approximating a 5-6-15-9 chord on 8/5!
>
>Besides a vague sound at 8/5, the 7/4 seems to get sharper as I go from 1/1
>5/4 3/2 7/4 to 1/1 6/5 3/2 7/4! Thinking maybe the dissonant 35/24 was
>throwing things off, I compared the 7/4 - 5/4 and the 7/4 - 6/5 dyads. And
>noticed no change in the 7/4.
>
>This is the most glaring example of an Erlich-type approximation I have
>ever heard. Can anyone confirm or deny?

Correct me if I'm wrong. You played the just chord
20:24:30:35, which contains the following intervals
5 6
4 5
4 6 7
2 3
24 35 (dissonance between 5:7 and 2:3)
in meantone terms say
F Ab C D#

and it sounded to you like an approximation of
10:12:15:18 (20:24:30:36)
5 6 9
4 5 6
2 3
2 3
F Ab C Eb

You heard a virtual fundamental (or was it a difference tone?) that was
octave equivalent to the lowest note divided by 5 (a Db). The highest note
seemed to go up in pitch relative to when it occurred in a 4:5:6:7, as you
might expect if its subjective relationship to the lowest note changed from
4:7 to 5:9.

I tried it and didn't hear either of these effects.

Note that 24:35 is 49 cents narrower than 2:3, way outside normal dyad
tolerance. What aspects of the various theories promoted by Paul Erlich do
you think would predict the results you heard?

20:24:30:35 is much closer to 16:19:24:28 than 10:12:15:18, for whatever
its worth.

Regards,
-- Dave Keenan
http://dkeenan.com

The post was messy, I agree.

>Correct me if I'm wrong. You played the just chord 20:24:30:35

Right.

>and it sounded to you like an approximation of 20:24:30:36

Yes!

>You heard a virtual fundamental (or was it a difference tone?) that was
>octave equivalent to the lowest note divided by 5 (a Db).

Well, sort of. It was more a sense than a hearing. You could say that of
all the pitches not sounding, it was the one I could most easily sing.

>The highest note seemed to go up in pitch relative to when it occurred in
a >4:5:6:7, as you might expect if its subjective relationship to the
lowest >note changed from 4:7 to 5:9.

Yes.

>I tried it and didn't hear either of these effects.

Fair enough. I tried it again, and I think I can still hear it. As I
mentioned, I do not get the pitch raise going from 25:35 to the 24:35. But
adding the 20 to each of these, I do hear it.

Some timbres don't work. I've lost my ftp capabilities, so I can't give
you an mp3. Try an electric piano. I used "percussive organ".

>Note that 24:35 is 49 cents narrower than 2:3, way outside normal dyad
>tolerance. What aspects of the various theories promoted by Paul Erlich do
>you think would predict the results you heard?

None. I mentioned Erlich's name just because it is associated with a type
of approximation. There are many types, you know. I mean the type you
said seemed dubious a few weeks ago.

>20:24:30:35 is much closer to 16:19:24:28 than 10:12:15:18, for whatever
>its worth.

Hmmm. Try a 16:20:24:30:36, and then try 16:20:24:30:35. Does the second
chord sound like a mistuning of the first?

C.

>Joseph L Monzo wrote:

>> [8/5] 1/1 5/4 3/2 9/5
>> [8:] 10 : 12 : 15 : 18
>> = [16:] 20 : 24 : 30 : 36
>>
>> [8/5] 1/1 5/4 3/2 7/4
>> [16:] 20 : 24 : 30 : 35

>Excuse me, is'nt it...

> [8/5] 1/1 5/4 3/2 9/5
> [16:] 20 : 25 : 30 : 36
>
> [8/5] 1/1 5/4 3/2 7/4
> [16:] 20 : 25 : 30 : 35

>David Mezquita.

Actually, it was 6/5, not 5/4, in the chords in question, so Joe got the
integers right.

Dave Keenan wrote,

>You heard a virtual fundamental (or was it a difference tone?) that was
>octave equivalent to the lowest note divided by 5 (a Db). The highest
note
>seemed to go up in pitch relative to when it occurred in a 4:5:6:7, as
you
>might expect if its subjective relationship to the lowest note changed
from
>4:7 to 5:9.

>I tried it and didn't hear either of these effects.

These effects are highly dependent on timbre. A timbre low in harmonics
would be best.

[me:]

>> [8/5] 1/1 5/4 3/2 9/5
>> [8:] 10 : 12 : 15 : 18
>> = [16:] 20 : 24 : 30 : 36
>>
>> [8/5] 1/1 5/4 3/2 7/4
>> [16:] 20 : 24 : 30 : 35

[David Mezquita:]

> Excuse me, is'nt it...
>
> [8/5] 1/1 5/4 3/2 9/5
> [16:] 20 : 25 : 30 : 36
>
> [8/5] 1/1 5/4 3/2 7/4
> [16:] 20 : 25 : 30 : 35

Oops! What a blunder!
Yes, David, your proportions are absolutely correct.

The mistake was that 5/4 should have been 6/5 in both
chords. In that case my proportions would be correct.
These are the chords Carl used in his experiment, and
how my original posting should have read:

[8/5] 1/1 6/5 3/2 9/5
[8:] 10 : 12 : 15 : 18
= [16:] 20 : 24 : 30 : 36

[8/5] 1/1 6/5 3/2 7/4
[16:] 20 : 24 : 30 : 35

(Dave Keenan has already posted to this effect)

BTW, I tried it and I hear it too! As Carl said,
I don't "hear" the (missing) fundamental, but the
7/4 definitely seems to sharpen to a higher note
when the 5/4 is changed to a 6/5.

- Monzo
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[Carl Lumma:]
>Well, sort of. It was more a sense than a hearing. You could say that of
>all the pitches not sounding, it was the one I could most easily sing.

Ok. I can go along with that.
20:24:30:35 has differences 4,6,5,10,11,15. All except 4 and 11 are octave
equivalent to notes of the chord. Can you sing an 11/10 as well as an 8/5?

>Fair enough. I tried it again, and I think I can still hear it. As I
>mentioned, I do not get the pitch raise going from 25:35 to the 24:35. But
>adding the 20 to each of these, I do hear it.

So you don't need the 30?

>Some timbres don't work. I've lost my ftp capabilities, so I can't give
>you an mp3. Try an electric piano. I used "percussive organ".

What's an mp3? I was using a sawtooth wave with an organ envelope (i.e.
continuous). Maybe that's the difference. A short attack-decay time gives
enough ambiguity in the frequency (the uncertainty principle).

Ok. I tried it.... Maybe the high note seems to go up.

>Hmmm. Try a 16:20:24:30:36, and then try 16:20:24:30:35. Does the second
>chord sound like a mistuning of the first?

Sort of. But shouldn't the second chord sound just as much like a mistuning
of 16:20:24:30:34 (8:10:12:15:17)? In this case the high note is flatter. I
tried that too (with a piano tone), and I'm willing to grant that maybe the
:35 sounds more like the :36 than the :34. But it's subtle. They mostly
just sound like 3 different chords to me, all fairly dissonant.

Certainly we could explain the preference for :36 by stepping down from
looking at the numbers in the whole chord to looking at the subchords and
intervals in lowest terms. There are many more small-numbered subsets in
the :36 chord than in the :34 chord.

These chords (:34, :35, :36) have Euler's Gradus Suavitatus (EGS) of 26,
17, 12 respectively. EGS does, in a sense, consider the subchords since it
takes the LCM. A reminder: EGS is: Find LCM/GCD, find its prime factors,
add up all the factors, subtract the number of factors, add one.

[A long time ago Paul Erlich wrote:]
The diminished seventh chord is tuned 10:12:14:17, and the dominant
flat-9 is tuned 8:10:12:14:17, when performed by a barbershop quartet or
other free-pitched ensemble.

Note the 14 in place of the 15 we've been considering. Don't know what this
means. Just thought I'd mention it. These have EGS's of 31 and 32
respectively.

-- Dave Keenan
http://dkeenan.com