High third experiment

From: Joe Monzo

> I've been intrigued by the observation Jerry Eskelin
> made that singers intuitively sing a 4:5 'major 3rd'
> [= ~386 cents] with the 1/1 'root' [= 0 cents] alone,
> but that when a 2:3 '5th' [= ~702 cents] is added to
> this dyad to make a complete 'major triad', the 'major 3rd'
> slides upward in pitch until it is larger than the
> 12-tET/12-EDO 'major 3rd' [= 2^(4/12) = 400 cents].

I'm delighted, Joe, that you have enlisted in the quest for the mysterious
third. This is the kind of thing I had hoped for when I got involved here.

> The 7:9 has been pretty much rejected as not one that is
> likely to be instinctively accepted as a smooth consonance.

Really??? I thought it sounded pretty good when John Link played "it"(?) on
his guitar over the phone. Are we talking about the same interval? Perhaps
John has a comment.
>
> I've made a MIDI file of some possibilities for this
> 'high 3rd', with some interesting results.
> http://www.ixpres.com/interval/td/monzo/high3rd.mid
>
> The chord is sequenced like the old Three Stooges
> 'hello hello hello' routine: in 4/4 time, the ratios
> 1/1 on the first beat, 5/4 on the third, and 3/2 on the
> first beat of the second measure, with pitch-bend applied
> continuously to the 5/4 between the first and second beats
> of the second measure to create a glissando up to the target
> 'high 3rd'.

It's a little difficult to evaluate the target tunings since these sustain
for only a short time. Would it be convenient to add a beat or two to each
item to let the target tuning sustain a bit longer? Does your software allow
for this without redoing the whole set up?

Also, I agree with Paul that it would help "fine tune" the experiment if the
vibrato could be eliminated. "Straight" tone is easier to tune (as well as
evaluate for tuning).

> Here are the possibilities I used:
>
> 1) m1-2 The 4:5 is held thru the entire chord.
> I agree with Jerry that it doesn't sound like what
> singers normally do.

I noticed an interesting thing while listening to this one. Although you did
not bend the third here, I had the impression that it raised slightly when
the fifth came in. Possibilities: (1) perhaps I "willed" it to move out of
habit, (2) perhaps the high third is simply a perceptual illusion, (3) and
if 2 is true, perhaps singers are influenced by the illusion and actually
attempt to sing what they have been "hearing."

I'm not quite ready to believe those "perhaps"'s at this point, but I'm
trying to keep an open mind.

> 2) m3-4 The 4:5 slides upward to 64:81.
> This was the one that I thought would have sounded
> the smoothest, based on the numerics of the ratios.
> Not bad, but (surprisingly) to my ears not the best;
> it sounds like it goes a little too sharp.
>
> 3) m5-6 The 4:5 slides upward to 2^(4/12).
> To me, surprisingly, this was a close second-best.
> Perhaps the role of cultural conditioning plays
> a stronger part than we realize.
>
> 4) m7-8 The 4:5 slides upward to 19:24.
> Most surprisingly of all, this possibility suggested
> by Paul Erlich turned out to be the smoothest to my ears.

2, 3, and 4 all sound "in the ball park" to me. I sang "my third" to each
item (from the beginning of the item) and in these cases the third came
close to what I was singing (give or take for vibrato).
>
> 5) m9-10 The 4:5 slides upward to 11:14.
> This one sounds pretty good to me, but a little too sharp.
>
> 6) m11-12 The 4:5 slides upward to 25:32.
> This one definitely sounds too sharp to me,
> which would certainly rule out 7:9.

The last two are so high they sound more like sus chords than major triads.
Certainly the 7:9 third, which sounds nothing like a perfect fourth, is not
as high as these.

There are 7 items on your sound example and only 6 described here. What did
I miss?
>
> I .... used the General MIDI 'Choir Ahs' patch.

Most synthesized "Choir" patches are doused in vibrato. If you can't iron it
out, you might try a flute, oboe or string patch (although string patches
frequently are pretty syruppy, too).

> Feedback definitely wanted.

Thanks again for your efforts here. Hope this "feedback" is helpful.

Jerry

P.S. I hope to post some sounds next week. My "new" college choir will be
the guinea pigs.

Joe Monzo wrote,

>> 5) m9-10 The 4:5 slides upward to 11:14.
>> This one sounds pretty good to me, but a little too sharp.
>
>> 6) m11-12 The 4:5 slides upward to 25:32.
>> This one definitely sounds too sharp to me,
>> which would certainly rule out 7:9.

Gerald Eskelin wrote,

>The last two are so high they sound more like sus chords than major triads.
>Certainly the 7:9 third, which sounds nothing like a perfect fourth, is not
>as high as these.

Sorry, Jerry, but a quick calculation shows:

11:14 = 417.5�
25:32 = 427.4�
7:9 = 435.1�

>Possibilities: (1) perhaps I "willed" it to move out of
>habit, (2) perhaps the high third is simply a perceptual illusion, (3) and
>if 2 is true, perhaps singers are influenced by the illusion and actually
>attempt to sing what they have been "hearing."

It is a psychoacoustically established fact that adding new tones can alter
the perceived pitch of constant, sustained tones. However, this effect is
normally demonstrated with sine waves, and I doubt it would continue to be
important with normal harmonic timbres. Terhardt, though, believes that our
inner mental template of the harmonic series is considerably "stretched" due
to this phenomenon.

Gerald Eskelin wrote,

>The last two are so high they sound more like sus chords than major triads.
>Certainly the 7:9 third, which sounds nothing like a perfect fourth, is not
>as high as these.

>There are 7 items on your sound example and only 6 described here. What did
>I miss?

This is what you missed: On Wed 1/19/00 12:41 PM, Joe Monzo wrote,

>I've added another rendition of the chord to the end of the
>MIDI file, with a glissando from 4:5 to 7:9.

>http://www.ixpres.com/interval/td/monzo/high3rd.mid

So it appears the two chords you thought sounded like a sus chord were in
fact the ones with 32:25 and 7:9 third. Vibrato aside, it seems, as I
suspected you would, that you've now judged the 7:9 major third as too high
for even the "high third". Furthermore, I'll venture a guess as to the
reason you thought these were sus chords: due to the thirds in the example
getting larger and larger, and the fact that you're not used to hearing
microtonal climbs in pitch, your ear at some point decided that the pitch
was a half-step above the major third, i.e., a perfect fourth. However, if
the example started with a 6:8:9 (sus chord) and then went down to 14:18:21
(chord with 7:9 major third), you would probably hear it as a major triad,
albeit still an out-of-tune one. Joe?

Anyway, Jerry, you evidently did not express an opinion about the fifth
chord, the one with a 11:14 major third. Given the amount of vibrato in the
example, I don't blame you. Still, I'm curious.

I said:
>
>>Possibilities: (1) perhaps I "willed" it to move out of
>>habit, (2) perhaps the high third is simply a perceptual illusion, (3) and
>>if 2 is true, perhaps singers are influenced by the illusion and actually
>>attempt to sing what they have been "hearing."

And Paul Erlich responded:
>
> It is a psychoacoustically established fact that adding new tones can alter
> the perceived pitch of constant, sustained tones. However, this effect is
> normally demonstrated with sine waves, and I doubt it would continue to be
> important with normal harmonic timbres. Terhardt, though, believes that our
> inner mental template of the harmonic series is considerably "stretched" due
> to this phenomenon.

This is extremely interesting, Paul. It seems very possible to me that I was
experiencing that phenomenon in this case. I'm not sure that it fully
explains why the high third appears to "lock," however, particularly since
you note that it appears to happen mainly with sine waves. On the other
hand, why do you "doubt it would continue to be important with normal
harmonic timbres"? Has this effect been studied with "normal" timbres and
been shown not to happen?

In any case, this opens a whole other world of possibilities.

BTW, You still haven't responded to my "off the wall" suggestion that the
shift in tuning may have to do with the combination of overtones (or
undertones) in the tonic and dominant fundamentals. Might that have any
relation to this discussion. (If that is a naive notion, just say so and
I'll forget it.)

Jerry

> Gerald Eskelin wrote,
>
>>The last two are so high they sound more like sus chords than major triads.
>>Certainly the 7:9 third, which sounds nothing like a perfect fourth, is not
>>as high as these.
>
> Sorry, Jerry, but a quick calculation shows:
>
> 11:14 = 417.5�
> 25:32 = 427.4�
> 7:9 = 435.1�

I hear (and see) you, Paul. But why do these last two (assuming they are
still 11:14 and 25:32 by the time I hear them via MIDI) sound like sus
chords to me? I wouldn't think of tuning a major third that high no matter
what else was sounding. (I haven't been able to get the added item
illustrating 7:9 to play. It must have downloaded corrupted. Since I have
trashed that list, perhaps it could be posted again?)

I just played the MIDI posted a few days ago (young9.mid) which sustains
7:8:9:12 and I sang a perfect fifth above 7 (14:21 as in the chord you
described earlier; 14:18:21, I believe) and the result sounded very much
like the singers' "high third" to me. Since John Link's guitar version also
sounded very "good" to me, I think the 7:9 theory moves to the top of the
list of possibilities.

But what's with the 417 and 427 above? Why do they sound more like a perfect
fourth than a major third? Something weird is going on here.

Jerry

Joe Monzo posted:

> I've prolonged each chord for an extra 4 beats, and
> changed the patch to General MIDI 'English Horn'. It
> still has vibrato, but less than any other appropriate
> sound I could find (i.e., excluding organ patches,
> which timbre would be totally inappropriate for this
> experiment concerning human voices).
> http://www.ixpres.com/interval/td/monzo/high3rd.mid

Thanks, Monz. I now have been able to open "high3rd2.mid" and agree that the
last item is very sharp. (I'm still confused about the number of items in
your various versions. Was the first one titled "high3rd.mid? If so, did it
include the 7:9 third? Actually, this is probably academic at this point.
Let's just drop it. I'll work on getting this one to play.
>
>> The last two are so high they sound more like sus chords
>> than major triads.
>
> Yes, I agree with that. Of course context plays a role
> here too. If the 1/1 - 32/25 - 3/2 chord were heard in
> a different context, with the '3rd' right on 32/25 rather
> than rising from the previous 5/4, it may sound more
> like a 'major' triad.

Yes, Paul mentioned that and I am keeping it in mind. Perhaps the solution
in this regard is to do a "blind test" of the different sonorities alone
(without the pitch bend showing) and see whether it results in a different
perception.

>> Certainly the 7:9 third, which sounds nothing like a
>> perfect fourth, is not as high as these.
>
> On the contrary, 7:9 [= ~435 cents] is noticeably higher
> (in this context, speaking of intervals, one should really
> say 'wider') than both 11:14 [= ~418 cents] and 25:32
> [= ~427 cents].
>
>> There are 7 items on your sound example and only 6 described here.
>> What did I miss?
>
> By the time you heard it, I had already updated it to
> include 4:5 rising to 7:9, as requested by John Link
> in TD 493.15.

With the same title (high3rd.mid)? I assume yes, since at that time I had
not yet heard "high3rd2.mid." On the other hand, "high3rd2.mid" also has
seven items. Am I confused??? Yes, I'm confused. But take heart in the fact
that it is not the first time I've been confused. :-)

Jerry

> [Jerry Eskelin, TD 499.22]
> BTW, You [Paul Erlich] still haven't responded to my
> "off the wall" suggestion that the shift in tuning may
> have to do with the combination of overtones (or undertones)
> in the tonic and dominant fundamentals. Might that have any
> relation to this discussion. (If that is a naive notion,
> just say so and I'll forget it.)

Actually, Jerry, Paul did respond to this idea.

> [Paul Erlich, TD 493.7]
> I don't think it's remotely possible for one to acoustically
> "lock" into the 81/64 over a 1/1 root and 3/2 fifth. That's
> why I suggested 24/19 -- it's still a long shot, but may be
> within the realm of possibility due to the common overtone.

This idea of 'sharing overtones' was also at the root of
my suggestion that 81/64 might be the 'high 3rd', but Paul
showed that I was fallaciously using prime-limit thinking
here.

> [Jerry Eskelin TD 499.8]
> I just played the MIDI posted a few days ago (young9.mid)
> which sustains 7:8:9:12 and I sang a perfect fifth above 7
> (14:21 as in the chord you described earlier; 14:18:21,
> I believe) and the result sounded very much like the singers'
> "high third" to me. Since John Link's guitar version also
> sounded very "good" to me, I think the 7:9 theory moves to
> the top of the list of possibilities.

Whoa, hold on there!

This file
http://www.ixpres.com/interval/td/monzo/young9.mid
actually contains the complete chord
mentioned by Daniel Wolf [TD 493.17], 32:48:56:63:72.
That's the 4:6:7:9 drone, with the 63/32 'pseudo-octave'
which is an 8:9 above 7/4, added in as the sung pitch.

With the vibrato your MIDI setup is producing, you probably
didn't notice the 'off' 32:63 'octave', hearing it instead
as 1:2, and apparently you're hearing the '6' as a '12'
(i.e., an 'octave' too high). Those two mishearings would
give the 7:8:9:12 you mention.

But the presence of 63/32 in the chord would certainly help
your sung 21/16 'lock', as can be readily seen from the
lattice:

63/32 ---------- 9/8
/ /
/ /
21/16 ---------- 3/2
/ /
/ /
7/4 ----------- 1/1

The 4:6:7:9 describes the pitches 1/1 - 3/2 - 7/4 - 9/8.

> [Jerry Eskelin, TD 499.27]
>> http://www.ixpres.com/interval/td/monzo/high3rd.mid
>
> Thanks, Monz. I now have been able to open "high3rd2.mid"
> and agree that the last item is very sharp. (I'm still
> confused about the number of items in your various versions.
> Was the first one titled "high3rd.mid? If so, did it
> include the 7:9 third? Actually, this is probably academic
> at this point. Let's just drop it. I'll work on getting
> this one to play.
> <snip>
> With the same title (high3rd.mid)? I assume yes, since at
> that time I had not yet heard "high3rd2.mid." On the other
> hand, "high3rd2.mid" also has seven items. Am I confused???
> Yes, I'm confused. But take heart in the fact that it is
> not the first time I've been confused. :-)

Now I'm confused! I never created a file named 'high3rd2.mid'!
I suppose it's possible that my FTP program added that '2'
without my noticing it.

The URL given above is the only filename I ever intended to
use for this experiment, and it should contain 7 different
versions of the 'high 3rd':
~cents
1. 5/4 all the way 386
2. 5/4 going to 81/64 408
3. 5/4 going to 2^(4/12) 400
4. 5/4 going to 24/19 404
5. 5/4 going to 14/11 418
6. 5/4 going to 32/25 427
7. 5/4 going to 9/7 435

If it doesn't sound like that, then I don't know what happened.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Jerry wrote,

>BTW, You still haven't responded to my "off the wall" suggestion that the
>shift in tuning may have to do with the combination of overtones (or
>undertones) in the tonic and dominant fundamentals. Might that have any
>relation to this discussion. (If that is a naive notion, just say so and
>I'll forget it.)

I did respond to that -- it was the first time I suggested 1/24:1/19:1/16.
The common overtone of these notes is the dominant, four octaves higher. As
for undertones, the "undertone series" does not occur in nature as a
simultaneous event, the way the overtone series does.

>From: PERLICH@ACADIAN-ASSET.COM
>
>Jerry wrote,
>
>>BTW, You still haven't responded to my "off the wall" suggestion that the
>>shift in tuning may have to do with the combination of overtones (or
>>undertones) in the tonic and dominant fundamentals. Might that have any
>>relation to this discussion. (If that is a naive notion, just say so and
>>I'll forget it.)
>
>I did respond to that -- it was the first time I suggested 1/24:1/19:1/16.
>The common overtone of these notes is the dominant, four octaves higher.

In this example the least common overtone occurs at the 24th, 19th, and
16th overtones of the tones in the triad. On the other hand, the three
tones in the major triad 1/9:1/7:1/6 (which is made of a major third at 9/7
and a fifth at 3/2, what Jerry wrote is still his best guess of the triad
with the high third), have their least common overtone at the 9th, 7th, and
6th overtones. We can write the least common overtone (LCO) as follows:

LCO = 6 * (1/6) = [(2*2)*(3/2)] * (1/6)

So the least common overtone is two octaves and a fifth above the fifth of
the triad (compare to four octaves above the fifth in the 1/24:1/19:1/16
triad). If we're talking about a C triad then the least common overtone is
a D.

The numbers 9, 7, and 6 being lower than 24, 19, and 16, I believe that it
would be easier to lock at 1/9:1/7:1/6 than at 1/24:1/19:1/16.

Let's take this utonal major triad 1/9:1/7:1/6 and see how we can extend
it. The obvious choices for tones to add are 1/8 and 1/5. 1/4 and 1/3 won't
be considered separately because 1/4 is an octave above 1/8, and 1/3 is an
octave above 1/6. Let's consider them one at a time. Let's continue to
think in terms of a C triad.

1/8: This tone is 9:8 relative C. I.e. 1/8 = (9/8)*(1/9). So 1/8 = D. If we
sing C, D, E we have a major second of 9:8 followed by a larger major
second of 8:7 (compare to 9:8 followed by a smaller 10:9). Taking this tone
up an octave and adding it to the triad we have C E G D with a 3:2 between
the G and the D.

1/5: This tone is 6:5 relative to G. I.e., 1/5 = (6/5)*(1/6). Relative to
C, 1/5 is 9/5. I.e., 1/5 = (9/5)*(1/9). So 1/5 = Bb. Thus 1/9:1/7:1/6:1/5
is a dominant 7th chord. Note that it is not a septimal dominant 7th chord
in the usual sense, because the third, rather than the seventh, involves 7.
What about the tritone of this chord?

Bb / E = (1/5)/(1/7) = 7/5

Compare to the tritone in a 4:5:6:7 dominant 7th chord:

Bb / E = 7/5

How about that? The sizes of the tritones are the same. In the utonal 7th
chord the third and seventh are both raised the same amount relative to the
third and seventh in the otonal chord.

If we add both D (at 1/4) and Bb to the triad we obtain a dominant 9th
chord tuned as 1/9:1/7:1/6:1/5:1/4. (For me it is easier to think about it
as 9/9:9/7:9/6:9/5:9/4, or better yet, 1:9/7:3/2:9/5:9/4) The least common
overtone of the tones of this dominant 7th chord is the same as that for
the 1/9:1/7:1/6 triad. That least common overtone is two octaves above the
ninth of the chord.

John Link

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featuring original compositions as well as arrangements of instrumental
music by Brahe and Taylor, Chick Corea, Miles Davis, Claude Debussy, Bill
Evans, Ennio and Andrea Morricone, Modeste Mussorgsky, Erik Satie, and Earl
Zindars.

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I wrote:

> Xavier Charles offered me generously his preparation to a thesis
> "Contribution � l'�tude des probl�mes de la gamme et de la justesse dans la
> musique occidentale, vers une autre th�orie des hauteurs?" ( Contribution
> the the study of the problems of the scale and 'being in tune' in Western
> music, toward another pitch theory?)

The french word "justesse" would be better translated by "pitch accuracy". I
wanted to use "just intonation", but that seems to close to "Just
Intonation".

--Wim

On 1/23/00 I sent the following message but have not yet received a single
response. In case anything went wrong with the transmission I'm sending it
again.

>From: PERLICH@ACADIAN-ASSET.COM
>
>Jerry wrote,
>
>>BTW, You still haven't responded to my "off the wall" suggestion that the
>>shift in tuning may have to do with the combination of overtones (or
>>undertones) in the tonic and dominant fundamentals. Might that have any
>>relation to this discussion. (If that is a naive notion, just say so and
>>I'll forget it.)
>
>I did respond to that -- it was the first time I suggested 1/24:1/19:1/16.
>The common overtone of these notes is the dominant, four octaves higher.

In this example the least common overtone occurs at the 24th, 19th, and
16th overtones of the tones in the triad. On the other hand, the three
tones in the major triad 1/9:1/7:1/6 (which is made of a major third at 9/7
and a fifth at 3/2, what Jerry wrote is still his best guess of the triad
with the high third), have their least common overtone at the 9th, 7th, and
6th overtones. We can write the least common overtone (LCO) as follows:

LCO = 6 * (1/6) = [(2*2)*(3/2)] * (1/6)

So the least common overtone is two octaves and a fifth above the fifth of
the triad (compare to four octaves above the fifth in the 1/24:1/19:1/16
triad). If we're talking about a C triad then the least common overtone is
a D.

The numbers 9, 7, and 6 being lower than 24, 19, and 16, I believe that it
would be easier to lock at 1/9:1/7:1/6 than at 1/24:1/19:1/16.

Let's take this utonal major triad 1/9:1/7:1/6 and see how we can extend
it. The obvious choices for tones to add are 1/8 and 1/5. 1/4 and 1/3 won't
be considered separately because 1/4 is an octave above 1/8, and 1/3 is an
octave above 1/6. Let's consider them one at a time. Let's continue to
think in terms of a C triad.

1/8: This tone is 9:8 relative C. I.e. 1/8 = (9/8)*(1/9). So 1/8 = D. If we
sing C, D, E we have a major second of 9:8 followed by a larger major
second of 8:7 (compare to 9:8 followed by a smaller 10:9). Taking this tone
up an octave and adding it to the triad we have C E G D with a 3:2 between
the G and the D.

1/5: This tone is 6:5 relative to G. I.e., 1/5 = (6/5)*(1/6). Relative to
C, 1/5 is 9/5. I.e., 1/5 = (9/5)*(1/9). So 1/5 = Bb. Thus 1/9:1/7:1/6:1/5
is a dominant 7th chord. Note that it is not a septimal dominant 7th chord
in the usual sense, because the third, rather than the seventh, involves 7.
What about the tritone of this chord?

Bb / E = (1/5)/(1/7) = 7/5

Compare to the tritone in a 4:5:6:7 dominant 7th chord:

Bb / E = 7/5

How about that? The sizes of the tritones are the same. In the utonal 7th
chord the third and seventh are both raised the same amount relative to the
third and seventh in the otonal chord.

If we add both D (at 1/4) and Bb to the triad we obtain a dominant 9th
chord tuned as 1/9:1/7:1/6:1/5:1/4. (For me it is easier to think about it
as 9/9:9/7:9/6:9/5:9/4, or better yet, 1:9/7:3/2:9/5:9/4) The least common
overtone of the tones of this dominant 7th chord is the same as that for
the 1/9:1/7:1/6 triad. That least common overtone is two octaves above the
ninth of the chord.

John Link

****************************************************************************

Watch for the CD "Live at Saint Peter's" by the JOHN LINK VOCAL QUINTET,
featuring original compositions as well as arrangements of instrumental
music by Brahe and Taylor, Chick Corea, Miles Davis, Claude Debussy, Bill
Evans, Ennio and Andrea Morricone, Modeste Mussorgsky, Erik Satie, and Earl
Zindars.

****************************************************************************

Check out WWW.DUESBERG.COM for information that could make the difference
between life and death for you or someone you know.

****************************************************************************

>The numbers 9, 7, and 6 being lower than 24, 19, and 16, I believe that it
>would be easier to lock at 1/9:1/7:1/6 than at 1/24:1/19:1/16.

Agreed. However, the 7:9 is too wide to be musically acceptable as a stable
major third over a root; so high, in fact, that it sounds quite a bit like a
perfect fourth (witness Jerry's reaction to the MIDI file -- he said the
high third he sung was in the range of the 400-408� major thirds -- he
clearly heard the file correctly since he identified the opening interval
correctly as the pure 4:5 major third).

The reason I suggested 1/24:1/19:1/16 is that Jerry said the high third is
_not_ used when just the root is present (in that case the low 4:5 third is
used), but _is_ used when the root and fifth are present. My reasoning
(admittedly far-fetched) was that, on some level, the common overtone is
actually heard as an independent voice, and therefore must be one of the
notes actually in the chord. If the root is C, the common overtone of 4:5 is
E, which is in the C-E dyad, but of 4:5:6 is B, which is not in the C-E-G
triad. For the 7:9 dyad and the 14:18:21 = 1/9:1/7:1/6 chord, the common
overtone is D, not in either chord. But for the 19:24 dyad and the
1/24:1/19:1/16 chord, the common overtone is G, which is not in the C-E dyad
but is in the C-E-G triad. Hence the rule, of having the common overtone
agree with the chord, would explain Jerry's observation that the major third
dyad is sung as 4:5, while in the triad the major third falls in the
400-408� range, which would agree with 19:24 = 404�.

I wrote,

>> It is a psychoacoustically established fact that adding new tones can
alter
>> the perceived pitch of constant, sustained tones. However, this effect is
>> normally demonstrated with sine waves, and I doubt it would continue to
be
>> important with normal harmonic timbres. Terhardt, though, believes that
our
>> inner mental template of the harmonic series is considerably "stretched"
due
>> to this phenomenon.

Jerry wrote,

>This is extremely interesting, Paul. It seems very possible to me that I
was
>experiencing that phenomenon in this case. I'm not sure that it fully
>explains why the high third appears to "lock," however, particularly since
>you note that it appears to happen mainly with sine waves. On the other
>hand, why do you "doubt it would continue to be important with normal
>harmonic timbres"? Has this effect been studied with "normal" timbres and
>been shown not to happen?

The reason I doubt it is because the overtones would normally provide the
brain with a great deal of information to use in what is known as
periodicity hearing -- the neural impulses "lock in phase" with the note,
thereby providing the brain with a very precise reading as to its frequency.
But the effect might not be eliminated entirely. Here are some papers for
you to study (perhaps someone near a library who speaks German can summarize
them for us):

Wallister, K. (1969), "�ber die Abh�ngigkeiten der Tonh�henempfindung von
Sinust�nen vom Schallpegel, von �berlagertem drosselndem St�rschall und von
der Darbietungsdauer," Acustica 21, 211-221.

Terhardt, E., and Fastl, H. (1971). "Zum Einfluss von St�rt�nen und
St�ger�uchen auf die Tonh�he von Sinust�nen," Acustica 25, 53-61.

>In any case, this opens a whole other world of possibilities.

Keep your mind open. It seems you've already decided what the "right" answer
is, and chosen to take a very inconsistent view of the evidence in order to
support your belief. That's a dangerous kind of thinking, precisely the kind
I get the impression you've tried to warn us against in your book, and yet
you're falling into it yourself.

Finally, if you don't believe (a select subset of) Monz's MIDI files, why
don't you walk over to your M-1 and tune up those chords yourself?

>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
>>The numbers 9, 7, and 6 being lower than 24, 19, and 16, I believe that it
>>would be easier to lock at 1/9:1/7:1/6 than at 1/24:1/19:1/16.
>
>Agreed. However, the 7:9 is too wide to be musically acceptable as a stable
>major third over a root; so high, in fact, that it sounds quite a bit like a
>perfect fourth (witness Jerry's reaction to the MIDI file -- he said the
>high third he sung was in the range of the 400-408� major thirds -- he
>clearly heard the file correctly since he identified the opening interval
>correctly as the pure 4:5 major third).

Jerry has said that the third I played for him on my guitar over the phone
sounds like his high third, not like a perfect fourth. I've explained in a
previous post that I'm convinced that I had correctly tuned a 9/7 third.
Jerry said that the MIDI file tuned to 9/7 sounded like a perfect fourth
and not at all like the third he heard on my guitar. So I'm not convinced
that he heard the MIDI file correctly.

>The reason I suggested 1/24:1/19:1/16 is that Jerry said the high third is
>_not_ used when just the root is present (in that case the low 4:5 third is
>used), but _is_ used when the root and fifth are present. My reasoning
>(admittedly far-fetched) was that, on some level, the common overtone is
>actually heard as an independent voice, and therefore must be one of the
>notes actually in the chord. If the root is C, the common overtone of 4:5 is
>E, which is in the C-E dyad, but of 4:5:6 is B, which is not in the C-E-G
>triad. For the 7:9 dyad and the 14:18:21 = 1/9:1/7:1/6 chord, the common
>overtone is D, not in either chord. But for the 19:24 dyad and the
>1/24:1/19:1/16 chord, the common overtone is G, which is not in the C-E dyad
>but is in the C-E-G triad. Hence the rule, of having the common overtone
>agree with the chord, would explain Jerry's observation that the major third
>dyad is sung as 4:5, while in the triad the major third falls in the
>400-408� range, which would agree with 19:24 = 404�.

Given that 1/24:1/19:1/16 has its least common overtone 4 octaves above the
fifth of the chord, and 1/9:1/7:1/6 has its least common overtone an
ocatave and a fifth above the above fifth, I would suggest altering your
suggestion to make it slightly less far-fetched by dropping the requirment
that the least common overtone be one of the pitches of the chord,
especially since Jerry said that the 9/7 I played for him seems to be the
best candidate he's heard.

John Link

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>Jerry has said that the third I played for him on my guitar over the phone
>sounds like his high third, not like a perfect fourth.

Context, context! I'm sure Jerry would hear 9/7 as a major third if Monz
created a MIDI file that descended to it from a 4/3 perfect fourth, rather
than ascending to it in a chain of microtonal steps. Gary Morrison has
spoken of these context-dependent effects before.

>Given that 1/24:1/19:1/16 has its least common overtone 4 octaves above the
>fifth of the chord, and 1/9:1/7:1/6 has its least common overtone an
>ocatave and a fifth above the above fifth, I would suggest altering your
>suggestion to make it slightly less far-fetched by dropping the requirment
>that the least common overtone be one of the pitches of the chord,

You haven't accounted for the fact that Jerry believes the low third (4:5)
is used over the root alone, while he says the high third is used over the
root and fifth. Admittedly, my explanation is specious, but I was trying to
accomodate Jerry's suggestion that there is some kind of locking going on
the high third that can only occur when the fifth is present, and was
assuming the high third is in the range he found it to be when listening to
Monz's MIDI file.

To the post:
>
>>The numbers 9, 7, and 6 being lower than 24, 19, and 16, I believe that it
>>would be easier to lock at 1/9:1/7:1/6 than at 1/24:1/19:1/16.

Paul Erlich responded:
>
> Agreed. However, the 7:9 is too wide to be musically acceptable as a stable
> major third over a root; so high, in fact, that it sounds quite a bit like a
> perfect fourth (witness Jerry's reaction to the MIDI file -- he said the
> high third he sung was in the range of the 400-408� major thirds -- he
> clearly heard the file correctly since he identified the opening interval
> correctly as the pure 4:5 major third).

But what about the 7:9 third in the post describing the 4:6:7:9 drone? That
third, as I stated earlier, was perfect for tuning the "high third" triad.
The alleged 7:9 third in the MIDI file was not close to locking. It sounded
horrid (as heard on my computer). I'm quite sure, as I remember, that I
could not have added a vocal fifth to that sound to complete a satisfying
triad. What was the difference, assuming that both contained a valid 7:9
third?
>
> The reason I suggested 1/24:1/19:1/16 is that Jerry said the high third is
> _not_ used when just the root is present (in that case the low 4:5 third is
> used), but _is_ used

("seems to be preferred")

> when the root and fifth are present. My reasoning
> (admittedly far-fetched) was that, on some level, the common overtone is
> actually heard

audibly? sensed? other?

> as an independent voice, and therefore must

why?

> be one of the
> notes actually in the chord. If the root is C, the common overtone of 4:5 is
> E, which is in the C-E dyad, but of 4:5:6 is B, which is not in the C-E-G
> triad. For the 7:9 dyad and the 14:18:21 = 1/9:1/7:1/6 chord, the common
> overtone is D, not in either chord. But for the 19:24 dyad and the
> 1/24:1/19:1/16 chord, the common overtone is G, which is not in the C-E dyad
> but is in the C-E-G triad. Hence the rule, of having the common overtone
> agree with the chord, would explain Jerry's observation that the major third
> dyad is sung as 4:5, while in the triad the major third falls in the
> 400-408� range, which would agree with 19:24 = 404�.

Interesting....but (as you say) somewhat farfetched....nevertheless,
admirable.

The subject is still open, I trust.

Jerry

I wrote:

>>> Bottom line: If that was the 7:9 third, it sounds very close
>>> to what I have been calling the "high third."
>
> Joe Monzo wrote,
>
>>OK. But if you're hearing 7:9 as the 'high 3rd' here, you
>>should also be hearing it as the 'high 3rd' in my MIDI-file
>>of your experiment, because it's exactly the same interval!
>
>>I suggest that the context is playing tricks on you here.

To which Paul Erlich appears to celebrate some sort of victory:
>
> You got him, Joe!!! Good work. Clearly there are various "high thirds" that
> Jerry wants depending on context -- as the seventh and ninth of a dominant
> chord, he wants 7:9, but between the root and third of a major triad, he
> finds 7:9 "way too high" and prefers something in the range of 400-408�.

Before admitting some sort of "defeat," I would like to hear Mr. Monzo's 7:9
third with NO context. If I can't sing a fifth with it that completes a
locked-in "high third" triad, you (and Joe, I presume) can break open the
bubbly. If I CAN sing such a fifth, I'll expect a bottle to arrive at my
address not later than five days after the post. LOL

Jerry

>> when the root and fifth are present. My reasoning
>> (admittedly far-fetched) was that, on some level, the common overtone is
>> actually heard

>audibly? sensed? other?

Vocal overtones at around the 24th partial (of choir-level voices) are
normally above, but close to, the lower limit of audibility. I haven't tried
this experiment with 1/24:1/19:1/16, but I have been able to listen for, and
then hear then with loud, striking clarity, the common overtone of simpler
utonal chords. If you've learned to hear out the overtones of a single tone
or voice (if not, you're in for a treat), this will be _especially_ easy.