traktus5 wrote:

> I don't know the current status of this debate, but have often seen
> it stated in the psychacoust lit. that tonal fusion in a complex tone
> works within a range of 'mistuings'...

We all know that certain properties hold for mistunings.
That's the whole point of temperament.  What you said is
that the precise tuning doesn't matter.  That's a very
different assertion and one you haven't backed up.


                          Graham

Hi Kelly-

>hi Carl.  One reference to this issue occurs in Richard Parncutt's
>important 1988 "Revision of Terhardt's Psychoacoustical Model of the
>Root(s) of a musical chord."  On p. 70, he discusses Terhtards use of
>subharmonics for a chord root model, and on the 'mistuning' of the
>natural 7th harmonic, he writes that 'this does not effect the model,
>as mistunings by up to half a semitone may still be perceived as
>belonging to the tone [citations...]

There's nothing I can see here against harmonic entropy.

>...and variations in tuning may occur in performance of diatonic
>music [citations..] without effecting the musics perceived tonal
>structure.

"Tonal structure" sounds to me like it almost could be referring
to functional harmony, which is outside of the scope of harmonic
entropy or any model of psychoacoustic dissonance.

>I don't know the current status of this debate, but have often seen
>it stated in the psychacoust lit. that tonal fusion in a complex tone
>works within a range of 'mistuings'...

"Tonal fusion" is a loaded term, isn't it?  One would have to
know what it means to comment.

-Carl

hi Carl.  One reference to this issue occurs in Richard Parncutt's
important 1988 "Revision of Terhardt's Psychoacoustical Model of the
Root(s) of a musical chord."  On p. 70, he discusses Terhtards use of
subharmonics for a chord root model, and on the 'mistuning' of the
natural 7th harmonic, he writes that 'this does not effect the model,
as mistunings by up to half a semitone may still be perceived as
belonging to the tone [citations...]...and variations in tuning may
occur in performance of diatonic music [citations..]without effecting
the musics perceived tonal structure.  Such varaitions are esay to
eplain if musical intervals are regarded as no more that pitch
distances (as in Terhards model) rather than as frequency ratios (eg,
as in the  theory of Boomsliter and Creel, 1961.)"

I don't know the current status of this debate, but have often seen
it stated in the psychacoust lit. that tonal fusion in a complex tone
works within a range of 'mistuings'...

-Kelly

he writes "  --- In harmonic_entropy@yahoogroups.com, Carl Lumma
<carl@...> wrote:
>
> At 12:56 PM 12/29/2007, you wrote:
> >Believe me, a top source.  I belive it's based, partly, on the
work of Bloomslitter and Creel.
>
> What is based on Boomlitter & Creel?
>
> >All you have to do is look at the main papers in psychoacoustics
to see that the exact tuning of the interval doesn't matter for
phenom. such as tonal fusion.
>
> I'm not familiar with such papers.  Can you cite some examples?
>
> -Carl
>

At 12:56 PM 12/29/2007, you wrote:
>Believe me, a top source.  I belive it's based, partly, on the work of
Bloomslitter and Creel.

What is based on Boomlitter & Creel?

>All you have to do is look at the main papers in psychoacoustics to see that
the exact tuning of the interval doesn't matter for phenom. such as tonal
fusion.

I'm not familiar with such papers.  Can you cite some examples?

-Carl

---

To: harmonic_entropy@yahoogroups.com
From: carl@...
Date: Sat, 29 Dec 2007 08:59:27 -0800
Subject: Re: [harmonic_entropy] interval as pitch space, not frequency

At 08:51 AM 12/29/2007, you wrote:
>A leading Psychoacoustician wrote
>
>"I had a look at "Harmonic Entropy" and was immediately put off by the
>idea that intervals are frequency ratios (which ratio? most intervals
>have two...) and Euler's idea of harmony, that in my view has no
>physiological basis"
>
>Most definitions of consonance do not require exact integer tunings,
>only approximations.
>
>-Kelly

Hi Kelly,

Who said this? It doesn't seem like they understand harmonic
entropy very well.

-Carl

At 08:51 AM 12/29/2007, you wrote:
>A leading Psychoacoustician wrote
>
>"I had a look at "Harmonic Entropy" and was immediately put off by the
>idea that intervals are frequency ratios (which ratio? most intervals
>have two...) and Euler's idea of harmony, that in my view has no
>physiological basis"
>
>Most definitions of consonance do not require exact integer tunings,
>only approximations.
>
>-Kelly

Hi Kelly,

Who said this?  It doesn't seem like they understand harmonic
entropy very well.

-Carl

A leading Psychoacoustician wrote

"I had a look at "Harmonic Entropy" and was immediately put off by the
idea that intervals are frequency ratios (which ratio? most intervals
have two...) and Euler's idea of harmony, that in my view has no
physiological basis"

Most definitions of consonance do not require exact integer tunings,
only approximations.

-Kelly

traktus5 wrote:
>
>
> hi. Does anyone know of graduate programs in USA where there is a
> speciality in topics on this list? thanks, KElly
>

You should certainly consider UC Santa Barbara now that Clarence Barlow
is teaching there.

DJW

hi.  Does anyone know of graduate programs in USA where there is a
speciality in topics on this list?  thanks, KElly

approx 28 minutes in

--- In harmonic_entropy@yahoogroups.com, "Glenn Freeman"
<glennf@...> wrote:
>
> final 30 minutes.
>
> Carl Lumma wrote:
>
> > At 10:03 AM 7/8/2006, you wrote:
> > >For those who missed it, you can listen anytime until July 15th
by
> > >clicking the below link ...
> > >
> > >http://rchrd.com/mfom/mfom.m3u
> >
> > What time does it start?
> >
> > -Carl
>

final 30 minutes.

Carl Lumma wrote:

> At 10:03 AM 7/8/2006, you wrote:
> >For those who missed it, you can listen anytime until July 15th by
> >clicking the below link ...
> >
> >http://rchrd.com/mfom/mfom.m3u
>
> What time does it start?
>
> -Carl

approximately the last 30 minutes.

--- In harmonic_entropy@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 10:03 AM 7/8/2006, you wrote:
> >For those who missed it, you can listen anytime until July 15th by
> >clicking the below link ...
> >
> >http://rchrd.com/mfom/mfom.m3u
>
> What time does it start?
>
> -Carl
>

At 10:03 AM 7/8/2006, you wrote:
>For those who missed it, you can listen anytime until July 15th by
>clicking the below link ...
>
>http://rchrd.com/mfom/mfom.m3u

What time does it start?

-Carl

The radio premiere of David Beardsley's (http://biink.com/db/index.htm)
30-minute 2004
string quartet "as beautiful as a crescent of a new moon on a cloudless spring
evening" as
performed by Christina Fong (http://christinafong.com) can be heard at 11pm PST
Friday
July 7th on KALW 91.7 FM's "Music from Other Minds" in San Francisco and also
via the
internet at http://rchrd.com/mfom.

David's piece was submitted for a project initiated on the "Why Patterns?"
discussion list.
"For Feldman" (http://www.amazon.com/gp/product/B000F3T3PK) is second in a
series
featuring previously unreleased works by emerging composers. This 96kHz|24bit
Audio
DVD (plays in any DVD player) contains 4 world premiere recordings and the only
available
release of 3 works by Morton Feldman.

One day after dinner, Morton was sitting at the kitchen table having a taste of
water,
thinking about a carpet or two, when electricity discontinued. Sun set and
awareness
gradually changed to wonder as he saw the beauty of the crescent of a new moon
on a
cloudless spring evening passing by the windows. Many years later, Morton awoke
on his
couch from an afternoon nap. Venus, the cat, leapt off the window where she was
sleeping
permitting sunlight to pass through a glass reclining on a table. "That is" he
thought, "as
beautiful as a crescent of a new moon on a cloudless spring evening".

This piece is tuned to the system known as Just Intonation, rational intervals
from the
harmonic series.

For those who missed it, you can listen anytime until July 15th by clicking the
below link ...

http://rchrd.com/mfom/mfom.m3u

Glenn Freeman wrote:

> The radio premiere of David Beardsley's (http://biink.com/db/index.htm)
30-minute 2004
> string quartet "as beautiful as a crescent of a new moon on a cloudless spring
evening" as
> performed by Christina Fong (http://christinafong.com) can be heard at 11pm
PST Friday
> July 7th on KALW 91.7 FM's "Music from Other Minds" in San Francisco and also
via the
> internet at http://rchrd.com/mfom

Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
>
[snip]
> >
> > Let's take your first example:
> > Sequence: 4, 5, 6
> > Double the middle term:
> > Sequence : 4, 10, 6
> > Rearrange: 4, 6, 10
> >  so that : 4 + 6 = 10
> > Is that what you mean?
>
> Yes.

Good!


> > More generally, if we call the middle element m,
> > and the common difference d, we have:
> > Sequence: (m-d), m, (m+d)
> > Double the middle term:
> > Sequence : (m-d), 2m, (m+d)
> > Rearrange: (m-d), (m+d), 2m
> >  so that : (m-d) + (m+d) = 2m
> >
> > Yes, the pattern holds for any values of m and d
> > that are integers.

Did you follow this reasoning?  Do you see
that it means that "double the middle term
in an arithmetic progression is always equal
to the sum of the two outside terms"?

In other words, we've proved that the pattern
you suspected really does hold - always.


> > As I said, an interesting pattern.  But I can't
> > help but ask myself: what does this have to do
> > with Harmonic Entropy?
>
> Chords of the type 3:7:10 have the coherant
> difference tones --which is pertinant to the
> list, yes?

Yes, I see.


> --- and are useful ways to arrange chords, and wanted
> to know how the math worked.

So, I hope it is clear now?


> Or maybe I just need a math tutor?

That'll be 5 bucks an answer, then ...! ;-)


> Anyway, can you, personally, separate the love of math
> (or numbers) from love of music?!

Oh yes, easily.  They both draw on my love of pattern,
true, but each has also other dimensions and aspects
that the other doesn't share.  And maths doesn't make
me dance!

Regards,
Yahya

Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
[snip]
> > > ...Are there any more rigourous psychoacoustical
> > > studies of this issue?
>
> Oh, I find your comments very rigorous!  I just thought
> there might be peer-reviewed research, backed up by
> laboratory studies, etc (like by acousticians such as
> Helmholtz, Parncut, etc), which bear on this question.

There might be; I don't know.


> > More rigorous than what?  My impressions?  Your
> > impressions?  Your analysis?
> >
Regards,
Yahya

Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
[snip]
> By 'height', I was referring to how, for example, the major third
> in the chord c-e-a is 12:15, whereas it's "height" in the chord
> d4-f#4-b4-e5 (spelled one way) is 36:45.  So, I still consider that
> the first major third has a height, in a manner of speaking of 3
> (though I realize 3/3 = 1), and the height of the second third is
> 9, in a manner of speaking.

What you call "height" would seem to be what
mathematicians call "Greatest Common Divisor"
or "GCD".


> You have to admit that, according to Paul's theories, the height
> of a chord in a series is related to its harmonic entropy, ...

??? I don't think I can admit what I don't understand ...!


> so I'm just trying to 'tabulate' the 'heights of the intervals.
>  How would you do it?
>
> thanks, Kelly
>
> > Hi Kelly,
> >
> > --- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
> > >
> > >
> > > > > > Do you think this could have any acoustical significance?
> > > > > > It's a very nice sounding chord!
> > >
> > > > > ... no, I don't think it guarantees an overall
> > > > > "nice" sound.
> > >
> > > But there is a correlation between the 'height' of the
> > > interval in the series (ie, eg, 5/4 x 3 = 15/12) and
> > > the difference tone, so there could be a connection...
> >
> >
> > You do *very* strange arithmetic! ;-)
> >
> > You often write things like:
> >  5/4 x 3 = 15/12
> >
> > This should be, instead,
> >  5/4 x 3/3 = 15/12
> >
> > since
> >  a) you can always multiply any number by 1 without
> >     changing it, and
> >  b) 2/2 = 3/3 = 4/4 = 5/5 = ... = 1 = n/n for every
> >     natural number n.
> >
> > These facts mean you cna legitimately multiply BOTH
> > top and bottom (numerator and denominator) of any
> > fraction by the same number.
> >
> > Enough of the maths ... what do you mean by "height"?

Regards,
Yahya

hi Yahya,

--- In harmonic_entropy@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
> >
> > Hello.  For chords such as 3:7:10, and 4:6:10, where the
> > difference tones land on the chord tones (ie, 10-7=3, 10-6-4;
> > Helmholtz commented on these, and I like them too!)...I had
> > a question.  This is probably axiomatic to the math adept
> > (but an exciting mystery to me!):what math principle is it
> > that accounts for the fact that if you take 3 adjacent
> > numbers in a sequence (eg, 4,5,6...or 3,5,7) and double the
> > middle one (to get 4,6,10...and 3,7,10), that you get 10-7=3,
> > 10-6=4?  (Don't laugh...)
>
> Not laughing, Kelly; it's an interesting pattern.
>
> Let's take your first example:
> Sequence: 4, 5, 6
> Double the middle term:
> Sequence : 4, 10, 6
> Rearrange: 4, 6, 10
>  so that : 4 + 6 = 10
> Is that what you mean?

Yes.

> More generally, if we call the middle element m,
> and the common difference d, we have:
> Sequence: (m-d), m, (m+d)
> Double the middle term:
> Sequence : (m-d), 2m, (m+d)
> Rearrange: (m-d), (m+d), 2m
>  so that : (m-d) + (m+d) = 2m
>
> Yes, the pattern holds for any values of m and d
> that are integers.
>
> As I said, an interesting pattern.  But I can't
> help but ask myself: what does this have to do
> with Harmonic Entropy?

Chords of the type 3:7:10 have the coherant difference tones --which
is pertinant to the list, yes?--- and are useful ways to arrange
chords, and wanted to know how the math worked.  Or maybe I just need
a math tutor?   Anyway, can you, personally, separate the love of math
(or numbers) from love of music?!

> I almost think you need to join - or start! - a
> list on numerology. ;-)
>
> Regards,
> Yahya
>

--- In harmonic_entropy@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
Hi Yayha,
> Hi again Kelly,
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
> [snip]
> > > The tetrad am : bm : cn : dn has the property
> > > that dn/am = n/m if and only if a = d. (Here
> > > juxtaposition means multiplication: am = a x m
> > > etc.)  m and n play the roles of your 9 and 20.
> > > That means a and b correspond to 4 and 5, and
> > > that c and d correspond to 3 and 4:
> > >  36 : 45 : 60 : 80
> > > = 4 x 9 : 5 x 9 : 3 x 20 : 4 x 20
> > >
> > > 45/36 = 5 x 9 / 4 x 9 = 5/4
> > >
> > > 80/60 = 4 x 20 / 3 x 20 = 4/3
> > >
> > > 80/36 = 4 x 20 / 4 x 9 = 20/9
>
> Errata: in the above, I should have written:
>  45/36 = (5 x 9) / (4 x 9) = 5/4
>  80/60 = (4 x 20) / (3 x 20) = 4/3
>  80/36 = (4 x 20) / (4 x 9) = 20/9
>
> according to the usual expression-writing rules.
>
>
> > > So any tetrad am : bm : cn : an has the same
> > > interesting property you [referred] to
> > > Eg 2x2 : 2x3 : 1x7 : 2x7
> > > = 4 : 6 : 7 : 14 is an example with a low limit.
> > > Another example, engineered from yours, is
> > >  4 x 7 : 5 x 7 : 3 x 17 : 4 x 17
> > > = 28 : 35 : 51 : 68.
> > > Or again,
> > >  3 x 7 : 5 x 7 : 2 x 18 : 3 x 18
> > > = 21 : 35 : 36 : 54.
> > > This latter example has a highly dissonant 35:36,
> > > so ...
> >
> > Hmmm...I'm not sure if I would include the latter two
> > chords in the same category as the one I cited...
>
> This was *exactly* my point in the comment marked ***
> below!
>
>
> > ...  And 4:6:7:14 has an octave in it, which may make
> > it 'too easy' to fit a pattern.
>
> You make it a bit hard to follow your thinking when you
> keep changing the rules! ;-)

Sorry.  You guys really keep me on my toes.

I just chose very small
> values for c and a, namely 1 and 2.  If you don't want
> octaves, you can of course bar any of a:b, b:c and c:a
> from having that ratio.
>
>
> > > > Do you think this could have any acoustical significance?
> > > > It's a very nice sounding chord!
> >
>
> ***
> > > ... no, I don't think it guarantees an overall
> > > "nice" sound.
> ***
>
> >
> > > > Also, speaking of 'bottom' and 'top' intervals: in a four
> > > > note chord, do intervals formed by adjacent notes (eg,
> > > > 5/4 and 4/3, from our chord) have any more prominance to
> > > > our hearing system than do intervals formed by non-
> > > > adjacent notes (eg, the 5/3 and 16/9 in the above chord)?
> > >
> *****
> > > It's my impression that middle voices and
> > > intervals are usually the hardest to hear.
> > > After the melody, most people pick up the
> > > bass.
> *****
> >
> > > Try this experiment: play a Cma7 chord:
> > > C E G B
> > > then alter it to Cmima7:
> > > C Eb G B
> > > and C #5 ma7:
> > > C E G# B.
> > > They all have a family resemblance, don't
> > > they?
> > >
> > > Now play Cdom7:
> > > C E G Bb
> > > and Cm7:
> > > C Eb G Bb.
> > > It's a different family, right?
> > >
> > > Wait! I can already hear the objection! ;-)
> > > "The two minor chords make one family,
> > > and the rest make another."  Well, yes.
> > > The third above the root is usually very
> > > salient in determining mood and mode.  But
> > > apart from the third (of whatever size, but
> > > clearly more than a second and less than a
> > > fourth) above the root, if present, the most
> > > salient interval seems to me to be usually
> > > the outside one.
> >
> > For me, with 'consonant' chords like the maj 7th chord,
>
> I find the major 7th chord very pleasantly dissonant.
> YMMV ...

Personally, I find the interval of a major seventh more consonant than
an octave, sort of a 'near octave'.  Perhaps tuning disprepancies are
more of an issue with octaves.

>
> > or dom 7th chord you site, I don't really hear the
> > inner intervals, ...
>
> This was the point I made at the note marked *****
> above.
>
>
> > ... presumably because they fall fairly nicely into
> > one series.
>
> Seems like a fair analysis.
>
>
> > ...  But with the c-e-g#-b, I strongly hear the g#
> > and b...
>
> ... presumably because they *don't* fall fairly nicely
> into one series?
>
>
> > ...Are there any more rigourous psychoacoustical
> > studies of this issue?

Oh, I find your comments very rigorous!  I just thought there might be
  peer-reviewed research, backed up by laboratory studies, etc (like by
acousticians such as Helmholtz, Parncut, etc), which bear on this
question.

> More rigorous than what?  My impressions?  Your
> impressions?  Your analysis?
>
> Regards,
> Yahya
>

--- In harmonic_entropy@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
hi Yahya - thanks for the math correction.  I'm terrible at it, though
love numbers...

By 'height', I was referring to how, for example, the major third in
the chord c-e-a is 12:15, whereas it's "height" in the chord
d4-f#4-b4-e5 (spelled one way) is 36:45.  So, I still consider that
the first major third has a height, in a manner of speaking of 3
(though I realize 3/3 = 1), and the height of the second third is 9,
in a manner of speaking.

You have to admit that, according to Paul's theories, the height of a
chord in a series is related to its harmonic entropy, so I'm just
trying to 'tabulate' the 'heights of the intervals.   How would you do
it?

thanks, Kelly

> Hi Kelly,
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
> >
> >
> > > > > Do you think this could have any acoustical significance?
> > > > > It's a very nice sounding chord!
> >
> > > > ... no, I don't think it guarantees an overall
> > > > "nice" sound.
> >
> > But there is a correlation between the 'height' of the
> > interval in the series (ie, eg, 5/4 x 3 = 15/12) and
> > the difference tone, so there could be a connection...
>
>
> You do *very* strange arithmetic! ;-)
>
> You often write things like:
>  5/4 x 3 = 15/12
>
> This should be, instead,
>  5/4 x 3/3 = 15/12
>
> since
>  a) you can always multiply any number by 1 without
>     changing it, and
>  b) 2/2 = 3/3 = 4/4 = 5/5 = ... = 1 = n/n for every
>     natural number n.
>
> These facts mean you cna legitimately multiply BOTH
> top and bottom (numerator and denominator) of any
> fraction by the same number.
>
> Enough of the maths ... what do you mean by "height"?
>
> Regards,
> Yahya
>

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
>
> Hello.  For chords such as 3:7:10, and 4:6:10, where the
> difference tones land on the chord tones (ie, 10-7=3, 10-6-4;
> Helmholtz commented on these, and I like them too!)...I had
> a question.  This is probably axiomatic to the math adept
> (but an exciting mystery to me!):what math principle is it
> that accounts for the fact that if you take 3 adjacent
> numbers in a sequence (eg, 4,5,6...or 3,5,7) and double the
> middle one (to get 4,6,10...and 3,7,10), that you get 10-7=3,
> 10-6=4?  (Don't laugh...)

Not laughing, Kelly; it's an interesting pattern.

Let's take your first example:
Sequence: 4, 5, 6
Double the middle term:
Sequence : 4, 10, 6
Rearrange: 4, 6, 10
  so that : 4 + 6 = 10
Is that what you mean?

More generally, if we call the middle element m,
and the common difference d, we have:
Sequence: (m-d), m, (m+d)
Double the middle term:
Sequence : (m-d), 2m, (m+d)
Rearrange: (m-d), (m+d), 2m
  so that : (m-d) + (m+d) = 2m

Yes, the pattern holds for any values of m and d
that are integers.

As I said, an interesting pattern.  But I can't
help but ask myself: what does this have to do
with Harmonic Entropy?

I almost think you need to join - or start! - a
list on numerology. ;-)

Regards,
Yahya

Hi again Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
[snip]
> > The tetrad am : bm : cn : dn has the property
> > that dn/am = n/m if and only if a = d. (Here
> > juxtaposition means multiplication: am = a x m
> > etc.)  m and n play the roles of your 9 and 20.
> > That means a and b correspond to 4 and 5, and
> > that c and d correspond to 3 and 4:
> >  36 : 45 : 60 : 80
> > = 4 x 9 : 5 x 9 : 3 x 20 : 4 x 20
> >
> > 45/36 = 5 x 9 / 4 x 9 = 5/4
> >
> > 80/60 = 4 x 20 / 3 x 20 = 4/3
> >
> > 80/36 = 4 x 20 / 4 x 9 = 20/9

Errata: in the above, I should have written:
  45/36 = (5 x 9) / (4 x 9) = 5/4
  80/60 = (4 x 20) / (3 x 20) = 4/3
  80/36 = (4 x 20) / (4 x 9) = 20/9

according to the usual expression-writing rules.


> > So any tetrad am : bm : cn : an has the same
> > interesting property you [referred] to
> > Eg 2x2 : 2x3 : 1x7 : 2x7
> > = 4 : 6 : 7 : 14 is an example with a low limit.
> > Another example, engineered from yours, is
> >  4 x 7 : 5 x 7 : 3 x 17 : 4 x 17
> > = 28 : 35 : 51 : 68.
> > Or again,
> >  3 x 7 : 5 x 7 : 2 x 18 : 3 x 18
> > = 21 : 35 : 36 : 54.
> > This latter example has a highly dissonant 35:36,
> > so ...
>
> Hmmm...I'm not sure if I would include the latter two
> chords in the same category as the one I cited...

This was *exactly* my point in the comment marked ***
below!


> ...  And 4:6:7:14 has an octave in it, which may make
> it 'too easy' to fit a pattern.

You make it a bit hard to follow your thinking when you
keep changing the rules! ;-)  I just chose very small
values for c and a, namely 1 and 2.  If you don't want
octaves, you can of course bar any of a:b, b:c and c:a
from having that ratio.


> > > Do you think this could have any acoustical significance?
> > > It's a very nice sounding chord!
>

***
> > ... no, I don't think it guarantees an overall
> > "nice" sound.
***

>
> > > Also, speaking of 'bottom' and 'top' intervals: in a four
> > > note chord, do intervals formed by adjacent notes (eg,
> > > 5/4 and 4/3, from our chord) have any more prominance to
> > > our hearing system than do intervals formed by non-
> > > adjacent notes (eg, the 5/3 and 16/9 in the above chord)?
> >
*****
> > It's my impression that middle voices and
> > intervals are usually the hardest to hear.
> > After the melody, most people pick up the
> > bass.
*****
>
> > Try this experiment: play a Cma7 chord:
> > C E G B
> > then alter it to Cmima7:
> > C Eb G B
> > and C #5 ma7:
> > C E G# B.
> > They all have a family resemblance, don't
> > they?
> >
> > Now play Cdom7:
> > C E G Bb
> > and Cm7:
> > C Eb G Bb.
> > It's a different family, right?
> >
> > Wait! I can already hear the objection! ;-)
> > "The two minor chords make one family,
> > and the rest make another."  Well, yes.
> > The third above the root is usually very
> > salient in determining mood and mode.  But
> > apart from the third (of whatever size, but
> > clearly more than a second and less than a
> > fourth) above the root, if present, the most
> > salient interval seems to me to be usually
> > the outside one.
>
> For me, with 'consonant' chords like the maj 7th chord,

I find the major 7th chord very pleasantly dissonant.
YMMV ...


> or dom 7th chord you site, I don't really hear the
> inner intervals, ...

This was the point I made at the note marked *****
above.


> ... presumably because they fall fairly nicely into
> one series.

Seems like a fair analysis.


> ...  But with the c-e-g#-b, I strongly hear the g#
> and b...

... presumably because they *don't* fall fairly nicely
into one series?


> ...Are there any more rigourous psychoacoustical
> studies of this issue?

More rigorous than what?  My impressions?  Your
impressions?  Your analysis?

Regards,
Yahya

Hi Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
>
>
> > > > Do you think this could have any acoustical significance?
> > > > It's a very nice sounding chord!
>
> > > ... no, I don't think it guarantees an overall
> > > "nice" sound.
>
> But there is a correlation between the 'height' of the
> interval in the series (ie, eg, 5/4 x 3 = 15/12) and
> the difference tone, so there could be a connection...


You do *very* strange arithmetic! ;-)

You often write things like:
  5/4 x 3 = 15/12

This should be, instead,
  5/4 x 3/3 = 15/12

since
  a) you can always multiply any number by 1 without
     changing it, and
  b) 2/2 = 3/3 = 4/4 = 5/5 = ... = 1 = n/n for every
     natural number n.

These facts mean you cna legitimately multiply BOTH
top and bottom (numerator and denominator) of any
fraction by the same number.

Enough of the maths ... what do you mean by "height"?

Regards,
Yahya

> > > Do you think this could have any acoustical significance?
> > > It's a very nice sounding chord!

> > ... no, I don't think it guarantees an overall
> > "nice" sound.

But there is a correlation between the 'height' of the interval in the
series (ie, eg, 5/4 x 3 = 15/12) and the difference tone, so there
could be a connection...

Hello Harmonic Entropy List.  5:16:21:25, ...another chord which, I
think, sounds interesting and unique, and which has an interesting
number feature.  Among the chords I use, I haven't found any other 4
note chord, relatively high in the series, where every interval -- but
for the outer interval -- is already  reduced.  I believe this is a
result of the chord being only relatively high in the series (ie,
above 16), with the chord tones 21 and 25 being only 'first order',
and not multiples of those numbers, such as a chord whose harmonic
series numbers are in the forties or fifties or beyond).  If height in
the series is a factor in dissonance, couldn't this be relavent?

Regards, Kelly

Hi Yahya

> > thanks for suggesting the
> > above spelling for the chord d4-f#-b4-E5.  I don't know
> > if it's significant, but the intervals of this chord
> > have a neat feature not shared by any other high-in-the
> > series 4 note chords which I have been able to find yet
> > (though math adept people could probably 'back engineer'
> > one from the number pattern...)
>
> Exactly! ;-)
>
>
> > To wit, the intervals, being 5/4, 4/3, 4/3, 5/3, 16/9,
> > and 20/9, have the interesting feature that the 'bottom'
> > interval, 5/4, is multiplied by 9 to 'be in the series'
> > at 45/36; and the 'top' interval b4-e5 (4/3) is
> > multiplied by 20 to 'be in the series' at 60:80.  And...
> > (drum roll...) the outer interval has the ratio 20/9.
> > (So I'm referring to the appearence of 20 and 9 in both
> > locations.)
>
> The tetrad am : bm : cn : dn has the property
> that dn/am = n/m if and only if a = d. (Here
> juxtaposition means multiplication: am = a x m
> etc.)  m and n play the roles of your 9 and 20.
> That means a and b correspond to 4 and 5, and
> that c and d correspond to 3 and 4:
>  36 : 45 : 60 : 80
> = 4 x 9 : 5 x 9 : 3 x 20 : 4 x 20
>
> 45/36 = 5 x 9 / 4 x 9 = 5/4
>
> 80/60 = 4 x 20 / 3 x 20 = 4/3
>
> 80/36 = 4 x 20 / 4 x 9 = 20/9
>
> So any tetrad am : bm : cn : an has the same
> interesting property you referered to
> Eg 2x2 : 2x3 : 1x7 : 2x7
> = 4 : 6 : 7 : 14 is an example with a low limit.
> Another example, engineered from yours, is
>  4 x 7 : 5 x 7 : 3 x 17 : 4 x 17
> = 28 : 35 : 51 : 68.
> Or again,
>  3 x 7 : 5 x 7 : 2 x 18 : 3 x 18
> = 21 : 35 : 36 : 54.
> This latter example has a highly dissonant 35:36,
> so ...

Hmmm...I'm not sure if I would include the latter two chords in the
same category as the one I cited.  And 4:6:7:14 has an octave in it,
which may make it 'too easy' to fit a pattern.

> > Do you think this could have any acoustical significance?
> > It's a very nice sounding chord!

> ... no, I don't think it guarantees an overall
> "nice" sound.


> > Also, speaking of 'bottom' and 'top' intervals: in a four
> > note chord, do intervals formed by adjacent notes (eg,
> > 5/4 and 4/3, from our chord) have any more prominance to
> > our hearing system than do intervals formed by non-
> > adjacent notes (eg, the 5/3 and 16/9 in the above chord)?
>
> It's my impression that middle voices and
> intervals are usually the hardest to hear.
> After the melody, most people pick up the
> bass.

> Try this experiment: play a Cma7 chord:
> C E G B
> then alter it to Cmima7:
> C Eb G B
> and C #5 ma7:
> C E G# B.
> They all have a family resemblance, don't
> they?
>
> Now play Cdom7:
> C E G Bb
> and Cm7:
> C Eb G Bb.
> It's a different family, right?
>
> Wait! I can already hear the objection! ;-)
> "The two minor chords make one family,
> and the rest make another."  Well, yes.
> The third above the root is usually very
> salient in determining mood and mode.  But
> apart from the third (of whatever size, but
> clearly more than a second and less than a
> fourth) above the root, if present, the most
> salient interval seems to me to be usually
> the outside one.

For me, with 'consonant' chords like the maj 7th chord, or dom 7th
chord you site, I don't really hear the inner intervals, presumably
because they fall fairly nicely into one series.  But with the c-e-g#-
b, I strongly hear the g# and b...Are there any more rigourous
psychoacoustical studies of this issue?

Thanks, Kelly

Hello.  For chords such as 3:7:10, and 4:6:10, where the difference
tones land on the chord tones (ie, 10-7=3, 10-6-4; Helmholtz commented
on these, and I like them too!)...I had a question.  This is probably
axiomatic to the math adept (but an exciting mystery to me!):what math
principle is it that accounts for the fact that if you take 3 adjacent
numbers in a sequence (eg, 4,5,6...or 3,5,7) and double the middle one
(to get 4,6,10...and 3,7,10), that you get 10-7=3, 10-6=4?  (Don't
laugh...)

A similiar question, is: take the chord 12:15:20.  You have 5/4 x 3 =
15:12, and 4/3 x 5 = 20/15.  Then, 15-12 equals the difference tone 3,
and 20/15 equals the differnece tone 5.  It just seems neat that 5/4 x
3 = 15/12 (the position of the interval in that particular chord), and
that the difference between the two numbers (15 and 12) is the same
number with which the lower fraction (5/4) was multiplied.  So why are
the differnce tone and the 'interval height' the same?

thanks, Kelly

Hi Kelly,

--- In harmonic_entropy@yahoogroups.com, Kelly wrote:
>
> hi Paul (for when you come back), or others.  I can't
> locate the old thread, but thanks for suggesting the
> above spelling for the chord d4-f#-b4-E5.  I don't know
> if it's significant, but the intervals of this chord
> have a neat feature not shared by any other high-in-the
> series 4 note chords which I have been able to find yet
> (though math adept people could probably 'back engineer'
> one from the number pattern...)

Exactly! ;-)


> To wit, the intervals, being 5/4, 4/3, 4/3, 5/3, 16/9,
> and 20/9, have the interesting feature that the 'bottom'
> interval, 5/4, is multiplied by 9 to 'be in the series'
> at 45/36; and the 'top' interval b4-e5 (4/3) is
> multiplied by 20 to 'be in the series' at 60:80.  And...
> (drum roll...) the outer interval has the ratio 20/9.
> (So I'm referring to the appearence of 20 and 9 in both
> locations.)

The tetrad am : bm : cn : dn has the property
that dn/am = n/m if and only if a = d. (Here
juxtaposition means multiplication: am = a x m
etc.)  m and n play the roles of your 9 and 20.
That means a and b correspond to 4 and 5, and
that c and d correspond to 3 and 4:
  36 : 45 : 60 : 80
= 4 x 9 : 5 x 9 : 3 x 20 : 4 x 20

45/36 = 5 x 9 / 4 x 9 = 5/4

80/60 = 4 x 20 / 3 x 20 = 4/3

80/36 = 4 x 20 / 4 x 9 = 20/9

So any tetrad am : bm : cn : an has the same
interesting property you referered to.
Eg 2x2 : 2x3 : 1x7 : 2x7
= 4 : 6 : 7 : 14 is an example with a low limit.
Another example, engineered from yours, is
  4 x 7 : 5 x 7 : 3 x 17 : 4 x 17
= 28 : 35 : 51 : 68.
Or again,
  3 x 7 : 5 x 7 : 2 x 18 : 3 x 18
= 21 : 35 : 36 : 54.
This latter example has a highly dissonant 35:36,
so ...

> Do you think this could have any acoustical significance?
> It's a very nice sounding chord!

... no, I don't think it guarantees an overall
"nice" sound.


> Also, speaking of 'bottom' and 'top' intervals: in a four
> note chord, do intervals formed by adjacent notes (eg,
> 5/4 and 4/3, from our chord) have any more prominance to
> our hearing system than do intervals formed by non-
> adjacent notes (eg, the 5/3 and 16/9 in the above chord)?

It's my impression that middle voices and
intervals are usually the hardest to hear.
After the melody, most people pick up the
bass.

Try this experiment: play a Cma7 chord:
C E G B
then alter it to Cmima7:
C Eb G B
and C #5 ma7:
C E G# B.
They all have a family resemblance, don't
they?

Now play Cdom7:
C E G Bb
and Cm7:
C Eb G Bb.
It's a different family, right?

Wait! I can already hear the objection! ;-)
"The two minor chords make one family,
and the rest make another."  Well, yes.
The third above the root is usually very
salient in determining mood and mode.  But
apart from the third (of whatever size, but
clearly more than a second and less than a
fourth) above the root, if present, the most
salient interval seems to me to be usually
the outside one.

Regards,
Yahya

hi Paul (for when you come back), or others.  I can't locate the old
thread, but thanks for suggesting the above spelling for the chord d4-
f#-b4-E5.  I don't know if it's significant, but the intervals of
this chord have a neat feature not shared by any other high-in-the
series 4 note chords which I have been able to find yet (though math
adept people could probably 'back engineer' one from the number
pattern...)

To wit, the intervals, being 5/4, 4/3, 4/3, 5/3, 16/9, and 20/9, have
the interesting feature that the 'bottom' interval, 5/4, is
multiplied by 9 to 'be in the series' at 45/36; and the 'top'
interval b4-e5 (4/3) is multiplied by 20 to 'be in the series' at
60:80.  And...(drum roll...) the outer interval has the ratio 20/9.
(So I'm referring to the appearence of 20 and 9 in both locations.)
Do you think this could have any acoustical significance?  It's a
very nice sounding chord!

Also, speaking of 'bottom' and 'top' intervals: in a four note chord,
do intervals formed by adjacent notes (eg, 5/4 and 4/3, from our
chord) have any more prominance to our hearing system than do
intervals formed by non-adjacent notes (eg, the 5/3 and 16/9 in the
above chord)?

cheers, Kelly