Yahoo Tuning Groups Ultimate Backup tuning [tuning] Flat ninth

On Mon, 13 Feb 2006 Keenan Pepper wrote:
>
> On 2/13/06, Lorenzo Frizzera ... wrote:
> > Hi.
> >
> > Sorry if my questions are not specific on tuning but I think the answers
> > have something to do with it...
> > Why a diatonic chord as Em7/b9 is more dissonant and less used than G7b9
> > which includes a non-diatonic interval as a diminished seventh? And what do
> > you think is the best ninth for a m7/b5 chord and why?
> >
> > lorenzo
>
> I think the reason the dominant 7b9 chord is more common than the m7b9
> chord in "common practice" music is not its intrinsic consonance, but
> its use in cadences. The m7b9 doesn't have a "leading tone" so it
> doesn't really "go" anywhere, whereas cadences with the dominant 7b9
> are almost a cliché.

Keenan, you have a point there; no, two.

> This is a terrible place to ask about common practice music theory
> though ...

Isn't it though? ;-)

> ... Case in point, I would say the best ninth for a m7b5 chord is
> a quarter-tone-flat 9th, giving the beautiful chord 5:6:7:9:11.

Depends on where you place the root, I think.
And certainly depends on what the general
tuning is. If it's JI, and you DO get the
exact ratios 5:6:7:9:11, it's certainly quite
harmonious; the timbre will affect which
combination tones predominate, though this
chord implies difference tones of 1:2:3:4:5:6
as well, which gives a full overtone series to
7 and all odd overtones to 11. If it's 12-EDO,
I find, for a Cm7b5 chord rooted on C4, the
major 9th (D6) more harmonious than either
the minor 9th (Db6) or the quarter-tone-flat
9th (D-6) - at least for a sampled piano tone.

However, when I play the same notes an octave
lower, the beats make all versions rather ugly
and I can't really pick between them.

Lorenzo, to try to answer your first question,
as to why the Em7b9 sounds less harmonious
than the G7b9: Suppose we sound these
chords based on E4 and G4, in the octave above
middle C, on the piano, in close position - that is,
built in thirds: EGBDF and GBDFAb.

My guess is that the first partials (octaves) of
the root and third (E4 and G4 for Em7b5, or G4
and B5 for G7B9) are fairly strong.

In the case of the Em7b9, these two notes are
E5 and G5, and I have a strong sense that they
form a cluster of four notes with the D5 and
F5 actually played: DEFG.

However, in the case of the G7b9, these two
notes are G5 and B6, and only the G5 forms a
cluster, of only three notes, with the F5 and
Ab6 actually played: FGAb.

The second cluster has fewer notes, and thus
fewer internal intervals (three instead of six).
I think this makes us work harder to sort out
the relationships.

If common practice had not conditioned us to
expect discords to resolve stepwise, we would
not expect the Ab in the G7b9 to resolve to G;
but we do, and that note is already anticipated,
by being present as the first partial of the root.

Should we likewise expect the F of the Em7b9
to resolve to the E? Yes, of course; however,
both the root and third are present as first
partials on E and G. We know the F could resolve
downwards by step to E, or upwards by step to G;
both are harmonious chord tones. The chord has
ambiguous harmonic tendencies.

On the whole, I think that our common-practice
background leads us to find the Em7b9 more
confusing and less likely to resolve satisfactorily
than the G7b9. It's not that it lacks a leading
tone; rather that it has one discordant tone that
could easily resolve in two different directions.
It confounds our expectations.

I also doubt that the particular tuning (meantone,
12-EDO, some well-tempering or other, or even JI)
would make much difference to (my analysis of)
this effect.

Regards,
Yahya

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🔗lorenzofrizzera <lorenzo.frizzera@cdmrovereto.it>

Hi Yahya.

This leads me to think in terms of distribution.
Infact if I put five notes at the maximum distance on a chromatic
circle in 12-et I get a 7b9 chord (or a 9 chord?), while m7/b9 or
maj7/b9 are less distributed.
But if this is the reason why, doing the same consideration on
triads, an augmented triad is worst than a major one?

Lorenzo

> On Mon, 13 Feb 2006 Keenan Pepper wrote:
> >
> > On 2/13/06, Lorenzo Frizzera ... wrote:
> > > Hi.
> > >
> > > Sorry if my questions are not specific on tuning but I think
the answers
> > > have something to do with it...
> > > Why a diatonic chord as Em7/b9 is more dissonant and less used
than G7b9
> > > which includes a non-diatonic interval as a diminished
seventh? And what do
> > > you think is the best ninth for a m7/b5 chord and why?
> > >
> > > lorenzo
> >
> > I think the reason the dominant 7b9 chord is more common than
the m7b9
> > chord in "common practice" music is not its intrinsic
consonance, but
> > its use in cadences. The m7b9 doesn't have a "leading tone" so it
> > doesn't really "go" anywhere, whereas cadences with the dominant
7b9
> > are almost a clichÃ©.
>
> Keenan, you have a point there; no, two.
>
>
> > This is a terrible place to ask about common practice music
theory
> > though ...
>
> Isn't it though? ;-)
>
>
> > ... Case in point, I would say the best ninth for a m7b5 chord is
> > a quarter-tone-flat 9th, giving the beautiful chord 5:6:7:9:11.
>
> Depends on where you place the root, I think.
> And certainly depends on what the general
> tuning is. If it's JI, and you DO get the
> exact ratios 5:6:7:9:11, it's certainly quite
> harmonious; the timbre will affect which
> combination tones predominate, though this
> chord implies difference tones of 1:2:3:4:5:6
> as well, which gives a full overtone series to
> 7 and all odd overtones to 11. If it's 12-EDO,
> I find, for a Cm7b5 chord rooted on C4, the
> major 9th (D6) more harmonious than either
> the minor 9th (Db6) or the quarter-tone-flat
> 9th (D-6) - at least for a sampled piano tone.
>
> However, when I play the same notes an octave
> lower, the beats make all versions rather ugly
> and I can't really pick between them.
>
>
> Lorenzo, to try to answer your first question,
> as to why the Em7b9 sounds less harmonious
> than the G7b9: Suppose we sound these
> chords based on E4 and G4, in the octave above
> middle C, on the piano, in close position - that is,
> built in thirds: EGBDF and GBDFAb.
>
> My guess is that the first partials (octaves) of
> the root and third (E4 and G4 for Em7b5, or G4
> and B5 for G7B9) are fairly strong.
>
> In the case of the Em7b9, these two notes are
> E5 and G5, and I have a strong sense that they
> form a cluster of four notes with the D5 and
> F5 actually played: DEFG.
>
> However, in the case of the G7b9, these two
> notes are G5 and B6, and only the G5 forms a
> cluster, of only three notes, with the F5 and
> Ab6 actually played: FGAb.
>
> The second cluster has fewer notes, and thus
> fewer internal intervals (three instead of six).
> I think this makes us work harder to sort out
> the relationships.
>
> If common practice had not conditioned us to
> expect discords to resolve stepwise, we would
> not expect the Ab in the G7b9 to resolve to G;
> but we do, and that note is already anticipated,
> by being present as the first partial of the root.
>
> Should we likewise expect the F of the Em7b9
> to resolve to the E? Yes, of course; however,
> both the root and third are present as first
> partials on E and G. We know the F could resolve
> downwards by step to E, or upwards by step to G;
> both are harmonious chord tones. The chord has
> ambiguous harmonic tendencies.
>
> On the whole, I think that our common-practice
> background leads us to find the Em7b9 more
> confusing and less likely to resolve satisfactorily
> than the G7b9. It's not that it lacks a leading
> tone; rather that it has one discordant tone that
> could easily resolve in two different directions.
> It confounds our expectations.
>
> I also doubt that the particular tuning (meantone,
> 12-EDO, some well-tempering or other, or even JI)
> would make much difference to (my analysis of)
> this effect.
>
> Regards,
> Yahya
>
> --
> No virus found in this outgoing message.
> Checked by AVG Free Edition.
> Version: 7.1.375 / Virus Database: 267.15.6/257 - Release Date:
10/2/06
>

🔗Hudson Lacerda <hfmlacerda@yahoo.com.br>

Joe escreveu:
> Well, a minor triad is slightly more dissonant than a major triad (in
> a 7th chord). Here's the intervals sorted by consonance, based on
> Kyle Gann's article on just intonation. Correct me if I'm wrong:
> > unison oct 5th 4th maj6th maj3rd min3rd min6th min7th maj2nd maj7th
> 1/1 2/1 3/2 4/3 5/3 5/4 6/5 8/5 9/5 9/8 15/8 > > min2nd tritone
> 16/15 45/32 > > Maybe that info will help you. As you can see, the tritone or flatted
> fifth is quite 'diabolic'.

I somewhat disagree about the tritone. It is acoustically less dissonant than the semitone, major 7th and minor 9th intervals. In a musical tonal context the tritone has a special function, and its "tension" is related to this function, not directly to its acoustical properties. The interval of 600 cents is nearly as dissonant than the interval of 300 cents for a pair of harmonic complex tones, according a study on "Consonance of Complex Tones and Its Calculation Method" by Kameoka & Kuriyagawa (JASA Vol 45 nr 6 1969, p.1460-1469 -- see figures 7 and 8 at page 1465).

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🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Hudson Lacerda <hfmlacerda@yahoo.com.br>

Gene Ward Smith escreveu:
> --- In tuning@yahoogroups.com, Hudson Lacerda <hfmlacerda@...> wrote:
> > >>"Consonance of Complex Tones and Its Calculation Method" by Kameoka & >>Kuriyagawa (JASA Vol 45 nr 6 1969, p.1460-1469 -- see figures 7 and
> > 8 at > >>page 1465).
> > > What is that calculation method?
> It seems to me a considerably complex method, taking in account the frequencies and absolute pressure levels of each partial of the tones involved. That (well refered) study was published as a couple of papers in the same volume of JASA in 1969 (the 1st one is ``Consonace Theory Part I: Consonance of Dyads''), thus it is likely that there are today many other ways to simulate listeners perception. Anyway, here is the Abstract of the paper I refered to:

<<<<
Consonance Theory Part II: Consonance of Complex Tones and Its Calculation Method

Akio Kameoka and Mamoru Kuriyagawa

Central Research Laboratory,
Tokyo Shibaura (Toshiba) Electric Co., Ltd., 1 Komukai Toshiba-cho, Kawasaki, Japan

A theory for calculating subjective dissonance of static complex tones has been established. The theory proposes a dissonance perception model that assumes that the mutual interactions between two components constitute an essential additive unit contributing to the dissonance. The model introduces a new concept of ``dissonance intensity'' in a certain process of dissonance perception and extends the ``power law''to the dissonance sensation, which is not clearly related to a certain physical value. Practical calculation procedures are described according to the experimental results of dyads in Part I. Theoretical calculation for various kinds of complex tones showed good agreements with psychological experiments. An application to chords of synthesized harmonic complex tones predicted great dependence of consonance characteristics on the harmonic structures, which are not taken in account in the conventional theory of harmony. It became clear that the fifth (2:3) is not always a consonant interval. A chord of two tones that consist of only odd harmonics, for example, shows much worse consonance at the fith (2:3) than at the major sixth (3:5) or some other frequency ratios. This was proved true by psychological expriments carried out in an other institute (Sensory Inspection Committee in the Japan Union of Scientists and Engineers) with a different method of scaling. Thus, the fact warns against making a mistake in appling the conventional theory of harmony to synthetic musical tones that can take variety in the harmonic structure. The theory next explains difference in quality reduction of reproduced sounds through a nonlinear audio instrument by the physical characteristics of input sources, and it provides a measure of evaluating nonlinear distortion.
>>>>

Best,
Hudson

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🔗Kraig Grady <kraiggrady@anaphoria.com>

I am disappointed that they do not address even triads which is where all the fun starts to happen. Much which below that level seems too ambiguous to conclude much.
we know about timbre effect on dissonance but maybe there i something here but can't tell by the below. What is to be said of all the music this is not based on consonant/dissonant qualities acknowledging it application would be of cultural interest .
translation might be a problem.

<<<<
Consonance Theory Part II: Consonance of Complex Tones and Its Calculation Method

Akio Kameoka and Mamoru Kuriyagawa

Central Research Laboratory,
Tokyo Shibaura (Toshiba) Electric Co., Ltd., 1 Komukai Toshiba-cho, Kawasaki, Japan

>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗harold_fortuin <harold_fortuin@yahoo.com>

A few thoughts:

Keenan, I wish we had more discussion in the tuning forums about
"common practice" theory, at least in relation to various tuning
systems. One thing practicing/aspiring composers/improvisers should
keep in mind is that it is THEORY - and in fact it has big holes in
it, including issues like chord inversion (anyone like the tonic
as the final chord of a piece in 2nd inversion?), and the related
concept of octave equivalence.

I however do agree with Keenan and Hudson in thinking of the tritone
along the lines of a 7:5. Personally, I think in "common practice"
musical contexts that our ears bend intervals to their nearest,
simplest JI equivalents - and 7/5 is much simpler than 45/32! Along
similar lines, I think we hear the 7th in the dominant
7th chord (like G7) as 7/4, and in a minor 7th chord like 9/5 (as in
Gm7). And the minor 9th can be understood as the 17th! (I'm not found
of thinking of intervals as "5-limit", "7-limit", etc. - omit
multiplications and seek your answers directly
in the harmonic series first!)

But to more directly address the issue: Lorenzo, you might try
theorizing about the role you're envisioning for the chord - is it a
substitute for a dominant 7th in a cadence? Or does it perhaps occur
in the midst of a chromatic passage?

Since it sounds like you have some grasp of music theory, you may
wish to review some common practice concepts like "dominant" "tonic",
etc., especially in comparison to a range of music including the
classical masters and strong jazz and pop composers, to
learn more about chord usage. And as you study, try out these
yourself, and see where things lead you!

And remember that music is very "fractal" - concepts like "passing
tone" can be applied to chords and even entire phrases of music.

--- In tuning@yahoogroups.com, "lorenzofrizzera"
<lorenzo.frizzera@...> wrote:
>
> Hi Yahya.
>
> This leads me to think in terms of distribution.
> Infact if I put five notes at the maximum distance on a chromatic
> circle in 12-et I get a 7b9 chord (or a 9 chord?), while m7/b9 or
> maj7/b9 are less distributed.
> But if this is the reason why, doing the same consideration on
> triads, an augmented triad is worst than a major one?
>
> Lorenzo
>
>
> > On Mon, 13 Feb 2006 Keenan Pepper wrote:
> > >
> > > On 2/13/06, Lorenzo Frizzera ... wrote:
> > > > Hi.
> > > >
> > > > Sorry if my questions are not specific on tuning but I think
> the answers
> > > > have something to do with it...
> > > > Why a diatonic chord as Em7/b9 is more dissonant and less
used
> than G7b9
> > > > which includes a non-diatonic interval as a diminished
> seventh? And what do
> > > > you think is the best ninth for a m7/b5 chord and why?
> > > >
> > > > lorenzo
> > >
> > > I think the reason the dominant 7b9 chord is more common than
> the m7b9
> > > chord in "common practice" music is not its intrinsic
> consonance, but
> > > its use in cadences. The m7b9 doesn't have a "leading tone" so
it
> > > doesn't really "go" anywhere, whereas cadences with the
dominant
> 7b9
> > > are almost a clichÃ©.
> >
> > Keenan, you have a point there; no, two.
> >
> >
> > > This is a terrible place to ask about common practice music
> theory
> > > though ...
> >
> > Isn't it though? ;-)
> >
> >
> > > ... Case in point, I would say the best ninth for a m7b5 chord
is
> > > a quarter-tone-flat 9th, giving the beautiful chord 5:6:7:9:11.
> >
> > Depends on where you place the root, I think.
> > And certainly depends on what the general
> > tuning is. If it's JI, and you DO get the
> > exact ratios 5:6:7:9:11, it's certainly quite
> > harmonious; the timbre will affect which
> > combination tones predominate, though this
> > chord implies difference tones of 1:2:3:4:5:6
> > as well, which gives a full overtone series to
> > 7 and all odd overtones to 11. If it's 12-EDO,
> > I find, for a Cm7b5 chord rooted on C4, the
> > major 9th (D6) more harmonious than either
> > the minor 9th (Db6) or the quarter-tone-flat
> > 9th (D-6) - at least for a sampled piano tone.
> >
> > However, when I play the same notes an octave
> > lower, the beats make all versions rather ugly
> > and I can't really pick between them.
> >
> >
> > Lorenzo, to try to answer your first question,
> > as to why the Em7b9 sounds less harmonious
> > than the G7b9: Suppose we sound these
> > chords based on E4 and G4, in the octave above
> > middle C, on the piano, in close position - that is,
> > built in thirds: EGBDF and GBDFAb.
> >
> > My guess is that the first partials (octaves) of
> > the root and third (E4 and G4 for Em7b5, or G4
> > and B5 for G7B9) are fairly strong.
> >
> > In the case of the Em7b9, these two notes are
> > E5 and G5, and I have a strong sense that they
> > form a cluster of four notes with the D5 and
> > F5 actually played: DEFG.
> >
> > However, in the case of the G7b9, these two
> > notes are G5 and B6, and only the G5 forms a
> > cluster, of only three notes, with the F5 and
> > Ab6 actually played: FGAb.
> >
> > The second cluster has fewer notes, and thus
> > fewer internal intervals (three instead of six).
> > I think this makes us work harder to sort out
> > the relationships.
> >
> > If common practice had not conditioned us to
> > expect discords to resolve stepwise, we would
> > not expect the Ab in the G7b9 to resolve to G;
> > but we do, and that note is already anticipated,
> > by being present as the first partial of the root.
> >
> > Should we likewise expect the F of the Em7b9
> > to resolve to the E? Yes, of course; however,
> > both the root and third are present as first
> > partials on E and G. We know the F could resolve
> > downwards by step to E, or upwards by step to G;
> > both are harmonious chord tones. The chord has
> > ambiguous harmonic tendencies.
> >
> > On the whole, I think that our common-practice
> > background leads us to find the Em7b9 more
> > confusing and less likely to resolve satisfactorily
> > than the G7b9. It's not that it lacks a leading
> > tone; rather that it has one discordant tone that
> > could easily resolve in two different directions.
> > It confounds our expectations.
> >
> > I also doubt that the particular tuning (meantone,
> > 12-EDO, some well-tempering or other, or even JI)
> > would make much difference to (my analysis of)
> > this effect.
> >
> > Regards,
> > Yahya
> >
> > --
> > No virus found in this outgoing message.
> > Checked by AVG Free Edition.
> > Version: 7.1.375 / Virus Database: 267.15.6/257 - Release Date:
> 10/2/06
> >
>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗harold_fortuin <harold_fortuin@yahoo.com>

🔗Tom Dent <stringph@gmail.com>

Hmmm... yes it does - the horns are in E and sound a minor 6th lower
than their notated pitch, so the 2nd horn's E is lower than the
bassoon's A.

This doesn't exclude the possibility that a choir arrangement would be
unconvincing. The orchestration is quite subtle and creates a
'floating' effect where it is not absolutely certain which note is the
root. To me the point is to connect the movement to the other parts
of the symphony.

~~~T~~~

--- In tuning@yahoogroups.com, "harold_fortuin" <harold_fortuin@...>
wrote:
>
> Kraig,
>
> I found this online score of Beethoven's 7th:
> http://www.dlib.indiana.edu/variations/scores/akx3424/large/index.html
> and didn't find this to be the case.
>
> But I do think I've heard some women's choral arrangement(s) that
> ended with a tonic 6-4 chord, but I also recall this sounding
> unconvincing to me, at least for common practice harmony.
>
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@> wrote:
> >
> > Beethoven Sym no. 7 2nd movement begins and ends on a 6/4
> chord, .But this is influenced by melodic and neighboring movement
> keys
> > From: "harold_fortuin"
> >
> >
> > A few thoughts:
> >
> > including issues like chord inversion (anyone like the tonic
> > as the final chord of a piece in 2nd inversion?), and the related
> > concept of octave equivalence
> >

🔗harold_fortuin <harold_fortuin@yahoo.com>

I should've also listened to the beginning to remind myself of the
transposition of those horns being down and not up - so on that point
I stand corrected. Of course the chord fades into a root position A
major chord, but remember my point was that the 6-4 chord isn't used
as the final chord of a cadence.

But now for a look at the end of the movement, specifically the last
3 bars. We once again have the wind voicing of a 6-4 tonic A major as
in the beginning, but note this time that the chord starts along with
an A in the cellos and basses, and this A is certainly lower than the
horns. And note that the chord has a big decrescendo. The effect to
my ear is that this brief low A provides the sensation of a root
position chord through the end of the movement, since no further bass
motion occurs until the next movement.

I'd still agree that because the bass tone is brief, the lingering
wind harmony, especially as a 6-4, and even with the descrescendo,
does undermine the finality of the tonic chord, which of course does
provide the sense of a need for additional music - which of course
Beethoven provided.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
>
> Hmmm... yes it does - the horns are in E and sound a minor 6th lower
> than their notated pitch, so the 2nd horn's E is lower than the
> bassoon's A.
>
> This doesn't exclude the possibility that a choir arrangement would
be
> unconvincing. The orchestration is quite subtle and creates a
> 'floating' effect where it is not absolutely certain which note is
the
> root. To me the point is to connect the movement to the other parts
> of the symphony.
>
> ~~~T~~~
>
> --- In tuning@yahoogroups.com, "harold_fortuin" <harold_fortuin@>
> wrote:
> >
> > Kraig,
> >
> > I found this online score of Beethoven's 7th:
> >
http://www.dlib.indiana.edu/variations/scores/akx3424/large/index.html
> > and didn't find this to be the case.
> >
> > But I do think I've heard some women's choral arrangement(s) that
> > ended with a tonic 6-4 chord, but I also recall this sounding
> > unconvincing to me, at least for common practice harmony.
> >
> >
> > --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@> wrote:
> > >
> > > Beethoven Sym no. 7 2nd movement begins and ends on a 6/4
> > chord, .But this is influenced by melodic and neighboring
movement
> > keys
> > > From: "harold_fortuin"
> > >
> > >
> > > A few thoughts:
> > >
> > > including issues like chord inversion (anyone like the tonic
> > > as the final chord of a piece in 2nd inversion?), and the
related
> > > concept of octave equivalence
> > >
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> I am disappointed that they do not address even triads which is >
where all the fun starts to happen.

They were wise, in a strange sense, not to address triads because
their calculation methods begin to fail egregiously to agree with
observation when there are more than 2 or 3 notes in a chord. The
same problem afflicts Sethares's closely related formula.

> Much which below that level seems too ambiguous to conclude much.

> we know about timbre effect on dissonance but maybe there i
>something here but can't tell by the below.

Yes, this is purely partial-based, just like Sethares.

> What is to be said of all the music this is not based on
>consonant/dissonant qualities acknowledging it application would be
>of cultural interest .
> translation might be a problem.

>
>
>
>
> <<<<
> Consonance Theory Part II: Consonance of Complex Tones and Its
> Calculation Method
>
> Akio Kameoka and Mamoru Kuriyagawa
>
> Central Research Laboratory,
> Tokyo Shibaura (Toshiba) Electric Co., Ltd., 1 Komukai Toshiba-cho,
> Kawasaki, Japan
>
> A theory for calculating subjective dissonance of static complex
tones
> has been established. The theory proposes a dissonance perception
model
> that assumes that the mutual interactions between two components
> constitute an essential additive unit contributing to the
dissonance.
> The model introduces a new concept of ``dissonance intensity'' in a
> certain process of dissonance perception and extends the ``power
law''to
> the dissonance sensation, which is not clearly related to a certain
> physical value. Practical calculation procedures are described
according
> to the experimental results of dyads in Part I. Theoretical
calculation
> for various kinds of complex tones showed good agreements with
> psychological experiments. An application to chords of synthesized
> harmonic complex tones predicted great dependence of consonance
> characteristics on the harmonic structures, which are not taken in
> account in the conventional theory of harmony. It became clear that
the
> fifth (2:3) is not always a consonant interval. A chord of two
tones
> that consist of only odd harmonics, for example, shows much worse
> consonance at the fith (2:3) than at the major sixth (3:5) or some
other
> frequency ratios. This was proved true by psychological expriments
> carried out in an other institute (Sensory Inspection Committee in
the
> Japan Union of Scientists and Engineers) with a different method of
> scaling. Thus, the fact warns against making a mistake in appling
the
> conventional theory of harmony to synthetic musical tones that can
take
> variety in the harmonic structure. The theory next explains
difference
> in quality reduction of reproduced sounds through a nonlinear audio
> instrument by the physical characteristics of input sources, and it
> provides a measure of evaluating nonlinear distortion.
> >>>>
>
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
>

🔗Lorenzo Frizzera <lorenzo.frizzera@cdmrovereto.it>

Hi.

I've done some cut and paste of various answers.

>I think the reason the dominant 7b9 chord is more common than the m7b9
>chord in "common practice" music is not its intrinsic consonance, but
>its use in cadences. The m7b9 doesn't have a "leading tone" so it
>doesn't really "go" anywhere, whereas cadences with the dominant 7b9
>are almost a clichï¿½.

>It's not that it lacks a leading tone; rather that it has one discordant
>tone that
>could easily resolve in two different directions.
>It confounds our expectations. The chord has ambiguous harmonic tendencies.

I think the concept of "cadence" has to do with the root of each interval
which is based upon the harmonic series. In this sense would be a good idea
to consider Em7/b9 as an inversion of a G13 where the root fight hard
against the bass. But is not so easy to find the root of a chord.

>One reason is the virtual pitch phenomenon -- G7b9 can be readily
>heard as the 8th, 10th, 12th, 14th, and 17th harmonics of the
>fundamental, which becomes a clear harmonic root in this context,
>while Em7b9 has no simple harmonic-series interpretation and thus
>confuses the virtual pitch mechanism.

I don't know really well the virtual pitch mechanism. What chord should be more rooted between m7 (10:12:15:18) and m7/b5 (5:6:7:9)?

>the minor 9th can be understood as the 17th! (I'm not found
>of thinking of intervals as "5-limit", "7-limit", etc. - omit
>multiplications and seek your answers directly
>in the harmonic series first!)

If I would read the harmonic series as it is I would consider that a
diminished triad is more consonant than a minor one, which is not true in my
opinion. I think that it has to be considered any harmonic series of any
note in a chord which implies the concept of limit as a ramification.

>However, in the case of the G7b9, these two notes are G5 and B6, and only
>the G5 >forms a cluster, of only three notes, with the F5 and Ab6 actually
>played: FGAb.
>The second cluster (Em7/b9) has fewer notes, and thus fewer internal
>intervals (three >instead of six).

Correct me if I'm wrong: the dissonance of clusters comes from the
phenomenon of critical bandwidth. CB has a quite logarithmic dimension which
means that any interval minor than a minor third is perceived as dissonant.
If CB (and clusters) would be the only parameter of consonance, an augmented
triad of pure sine wave should be perceived more consonant than a major
triad. I doubt of it.
However the only tetrads without intervals of second is the diminished one
(C Eb Gb Bbb) and this can explain why it were so used and maybe (i don't
know) before other tetrads as dominant, maj7, m7 or 6 chords. De La Motte
wrote that Bach used it without thinking as a dominant ninth chord without
root. The distribution of the four notes in the octave could be the key of
this usage.
In this way a 7/b9 chord would be "more distributed" than a 9 chord because
it includes more minor thirds intervals and it would be the widest pentad in
12-et.

>a minor triad is slightly more dissonant than a major triad (in a 7th
>chord).

I'm not agree. I hear a m7 chord more consonant than a dominant chord.

lorenzo

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Lorenzo Frizzera" <lorenzo.frizzera@...> wrote:
>
> Hi.
>
> I've done some cut and paste of various answers.
>
>>
> I think the concept of "cadence" has to do with the root of each interval
> which is based upon the harmonic series.

I think the tritone and its melodic tendencies form a strong driving force for cadences in tonal diatonic music.

> In this sense would be a good idea
> to consider Em7/b9 as an inversion of a G13 where the root fight hard
> against the bass. But is not so easy to find the root of a chord.
>
> >One reason is the virtual pitch phenomenon -- G7b9 can be readily
> >heard as the 8th, 10th, 12th, 14th, and 17th harmonics of the
> >fundamental, which becomes a clear harmonic root in this context,
> >while Em7b9 has no simple harmonic-series interpretation and thus
> >confuses the virtual pitch mechanism.
>
> I don't know really well the virtual pitch mechanism. What chord should be
> more rooted between m7 (10:12:15:18) and m7/b5 (5:6:7:9)?

Should I take these ratios literally, and assume you are using at least adaptive JI?

> >the minor 9th can be understood as the 17th! (I'm not found
> >of thinking of intervals as "5-limit", "7-limit", etc. - omit
> >multiplications and seek your answers directly
> >in the harmonic series first!)

Well there's an important distinction between two definitions of limit, "odd-limit" and "prime-limit". The former is the one to use when looking at individual intervals, and does not allow much in the way of "multiplications"!

> If I would read the harmonic series as it is I would consider that a
> diminished triad is more consonant than a minor one, which is not true in my
> opinion.

Of course not. There are several other components of dissonance, such as roughness between partials that fall into the same critical band.

> I think that it has to be considered any harmonic series of any
> note in a chord which implies the concept of limit as a ramification.

Huh?

> >However, in the case of the G7b9, these two notes are G5 and B6, and only
> >the G5 >forms a cluster, of only three notes, with the F5 and Ab6 actually
> >played: FGAb.
> >The second cluster (Em7/b9) has fewer notes, and thus fewer internal
> >intervals (three >instead of six).
>
> Correct me if I'm wrong: the dissonance of clusters comes from the
> phenomenon of critical bandwidth. CB has a quite logarithmic dimension which
> means that any interval minor than a minor third is perceived as dissonant.

It's not that simple, except for sine waves. All the partials of the notes (often louder than the fundamental for real intervals) must be taken into account; all pairs of sounding partials forming part of the roughness calculation.

> If CB (and clusters) would be the only parameter of consonance, an augmented
> triad of pure sine wave should be perceived more consonant than a major
> triad. I doubt of it.

I share your doubt, mainly because of the virtual pitch phenomenon. Are you familiar with the harmonic entropy list?

> However the only tetrads without intervals of second is the diminished one
> (C Eb Gb Bbb)

That's not so -- consider a minor seventh chord. A minor seventh is *not* equivalent to a major second when it's critical band effects you're considering.

> and this can explain why it were so used and maybe (i don't
> know) before other tetrads as dominant, maj7, m7 or 6 chords.

Nope.

De La Motte
> wrote that Bach used it without thinking as a dominant ninth chord without
> root.

Right -- both of the tritonrs in the chord resolve in a tonally directed way, allowing for perhaps the strongest cadence in tonal music.

> In this way a 7/b9 chord would be "more distributed" than a 9 chord because
> it includes more minor thirds intervals and it would be the widest pentad in
> 12-et.

This doesn't seem to make sense to me. Can you elaborater this reasoning in a consistent fashion?

> >a minor triad is slightly more dissonant than a major triad (in a 7th
> >chord).
>
> I'm not agree. I hear a m7 chord more consonant than a dominant chord.

I tend to agree, but how you tune these chords can make a huge difference in the comparison.

> lorenzo
>

🔗Hudson Lacerda <hfmlacerda@yahoo.com.br>

wallyesterpaulrus escreveu:
[...]
>>>a minor triad is slightly more dissonant than a major triad (in a 7th
>>>chord).
>>
>>I'm not agree. I hear a m7 chord more consonant than a dominant chord.
> > > I tend to agree, but how you tune these chords can make a huge difference in the comparison.
[...]

Don't mess intervals with chords this way. The main difference in that example is not in the thirds, but in the fifths instead:

|====| perfect 5th
Cm7 : C Eb G Bb
|====| perfect 5th

|===| perfect 5th
C7 : C E G Bb
|====| *diminished 5th*

Non perfect 5ths have not the stability of a perfect 5th. Cm7 is more stable than C7, thus the consonance difference.

Hudson

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_______________________________________________________
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🔗Lorenzo Frizzera <lorenzo.frizzera@cdmrovereto.it>

> I think the tritone and its melodic tendencies form a strong driving force > for cadences in tonal diatonic music.

I'm agree. Probably a cadence derives from a mix between intervals roots and tritone movement. Anyway a simple V to I progression, which is the most common and maybe strongest cadence, derives only by intervals roots (the intervals between C and any note of V are rooted on C)

>> I don't know really well the virtual pitch mechanism. What chord should >> be
>> more rooted between m7 (10:12:15:18) and m7/b5 (5:6:7:9)?
>
> Should I take these ratios literally, and assume you are using at least > adaptive JI?

Yes

>> I think that it has to be considered any harmonic series of any
>> note in a chord which implies the concept of limit as a ramification.
>
> Huh?

i think the concept of limit is used to generalize from a single harmonic series to a group of linked harmonic series. As a tree.

>> If CB (and clusters) would be the only parameter of consonance, an >> augmented
>> triad of pure sine wave should be perceived more consonant than a major
>> triad. I doubt of it.
>
> I share your doubt, mainly because of the virtual pitch phenomenon. Are > you familiar with the harmonic entropy list?

I'm scared about it :-)
I don't think I'm ready to read that list. It's to difficult for me...
Anyway I'll try to do. First of all I'll have to read the definition of harmonic entropy on the encyclopedia of Joe Monzo.

>> However the only tetrads without intervals of second is the diminished >> one
>> (C Eb Gb Bbb)
>
> That's not so -- consider a minor seventh chord. A minor seventh is *not* > equivalent to a major second when it's critical band effects you're > considering.

I was just thinking in 12-et possibilities. This is the only four note chord without second intervals even doubling any note of it on higher or lower octaves.

> Right -- both of the tritonrs in the chord resolve in a tonally directed > way, allowing for perhaps the strongest cadence in tonal music.

I don't understand: B go to C and F go to E, as usual but while G# go to A, D go to C or E, which is a whole tone step. If I would consider this as a "tonally directed way" I have to consider any tonal progression as resolutive since any progression can be reducted to half or whole tone movements.

>> In this way a 7/b9 chord would be "more distributed" than a 9 chord >> because
>> it includes more minor thirds intervals and it would be the widest pentad >> in
>> 12-et.
>
> This doesn't seem to make sense to me. Can you elaborater this reasoning > in a consistent fashion?

We can consider consonant an interval only if greater than a minor third (I know it is very arguable). This has to do with the distance and the distribution between notes. From this point of view the 7/b9 chord is the widest pentad because there are just two intervals of second (one major and one minor) while the 9 chord has three of them (major).
I consider these second intervals in 12-et which is my daily contest. This thing together with circle of fifth gives me a quite convincing explanation of the progression of harmonic consonance:

C G D A E
without seconds: major and minor triad
with seconds: m7, add9, 6/9

C G D A E B
with seconds: maj7, m9, maj9, m11

C G D A E B F#
without seconds: diminished triad
with seconds: 7, m7/b5, any diatonic chord

Complete circle of fifths
without seconds: augmented triad, diminished tetrad
with second: any chromatic chord

I know it is not so scientifically based... but to my ears it works.

lorenzo

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Hudson Lacerda <hfmlacerda@...> wrote:
>
> wallyesterpaulrus escreveu:
> [...]
> >>>a minor triad is slightly more dissonant than a major triad (in a 7th
> >>>chord).
> >>
> >>I'm not agree. I hear a m7 chord more consonant than a dominant chord.
> >
> >
> > I tend to agree, but how you tune these chords can make a huge difference in the comparison.
> [...]
>
> Don't mess intervals with chords this way.

Who me?

> The main difference in that
> example is not in the thirds, but in the fifths instead:
>
> |====| perfect 5th
> Cm7 : C Eb G Bb
> |====| perfect 5th
>
> |===| perfect 5th
> C7 : C E G Bb
> |====| *diminished 5th*
>
> Non perfect 5ths have not the stability of a perfect 5th. Cm7 is more
> stable than C7, thus the consonance difference.
>
> Hudson

I agree that that's a huge part of it. But still, the comparison is *very* different if you use a pure 4:5:6:7 chord vs. using a 12-equal or meantone or Pythagorean or 1/9:1/7:1/6:1/5 chord. The first one has a far greater tendency to fuse into a single note in perception, making the individual intervals less relevant.

> --
> '-------------------------------------------------------------------.
> Hudson Lacerda <http://geocities.yahoo.com.br/hfmlacerda/>
> *Não deixe seu voto sumir! http://www.votoseguro.org/
> *Apóie o Manifesto: http://www.votoseguro.com/alertaprofessores/
>
> == THE WAR IN IRAQ COSTS ==
> http://nationalpriorities.org/index.php?option=com_wrapper&Itemid=182
> .-------------------------------------------------------------------'
> --
>
>
>
> _______________________________________________________
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> http://br.acesso.yahoo.com
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@...> wrote:
>
> > I think the tritone and its melodic tendencies form a strong
driving force
> > for cadences in tonal diatonic music.
>
> I'm agree. Probably a cadence derives from a mix between intervals
roots and
> tritone movement. Anyway a simple V to I progression, which is the
most
> common and maybe strongest cadence, derives only by intervals roots
(the
> intervals between C and any note of V are rooted on C)
>
> >> I don't know really well the virtual pitch mechanism. What chord
should
> >> be
> >> more rooted between m7 (10:12:15:18) and m7/b5 (5:6:7:9)?
> >
> > Should I take these ratios literally, and assume you are using at
least
> > adaptive JI?
>
> Yes
>
> >> I think that it has to be considered any harmonic series of any
> >> note in a chord which implies the concept of limit as a
ramification.
> >
> > Huh?
>
> i think the concept of limit is used to generalize from a single
harmonic
> series to a group of linked harmonic series. As a tree.

There are at least two distinct concepts of limit (odd and prime),
but I don't think your description applies perfectly to either one.
Both a single harmonic series segment, and a full diamond with enough
pitches to allow 1/1 to serve as every possible harmonic in this
segment as part of a full harmonic chord in each case, have the same
limit, according to both concepts, even though the latter has almost
the square of the number of pitches of the former.

> >> However the only tetrads without intervals of second is the
diminished
> >> one
> >> (C Eb Gb Bbb)
> >
> > That's not so -- consider a minor seventh chord. A minor seventh
is *not*
> > equivalent to a major second when it's critical band effects
you're
> > considering.
>
> I was just thinking in 12-et possibilities.

Whether 12-equal or not, my point remains true.

> This is the only four note chord
> without second intervals even doubling any note of it on higher or
lower
> octaves.

I got that that was your meaning.

> > Right -- both of the tritonrs in the chord resolve in a tonally
directed
> > way, allowing for perhaps the strongest cadence in tonal music.
>
> I don't understand: B go to C and F go to E, as usual but while G#
go to A,
> D go to C or E, which is a whole tone step.

I was taught that ideally, D should go to C, so that each tritone
resolves in contrary motion.

> If I would consider this as a
> "tonally directed way" I have to consider any tonal progression as
> resolutive since any progression can be reducted to half or whole
tone
> movements.

I was referring to the characteristic dissonance resolving in
contrary motion to notes of the tonic chord. I don't think that just
*any* tonal progression displays this behavior.

> >> In this way a 7/b9 chord would be "more distributed" than a 9
chord
> >> because
> >> it includes more minor thirds intervals and it would be the
widest pentad
> >> in
> >> 12-et.
> >
> > This doesn't seem to make sense to me. Can you elaborater this
reasoning
> > in a consistent fashion?
>
> We can consider consonant an interval only if greater than a minor
third (I
> know it is very arguable). This has to do with the distance and the
> distribution between notes. From this point of view the 7/b9 chord
is the
> widest pentad because there are just two intervals of second (one
major and
> one minor) while the 9 chord has three of them (major).

But I could argue that no major second dyad has as much critical-band
roughness as a triad in which all notes fall into a minor third span,
especially as you are ignoring overtones.

> I consider these second intervals in 12-et which is my daily
contest. This
> thing together with circle of fifth gives me a quite convincing
explanation
> of the progression of harmonic consonance:
>
> C G D A E
> without seconds: major and minor triad
> with seconds: m7, add9, 6/9

How about sus2 and sus4 chords? How about 7th chords with no 3rd?

> C G D A E B
> with seconds: maj7, m9, maj9, m11
>
> C G D A E B F#
> without seconds: diminished triad
> with seconds: 7, m7/b5, any diatonic chord
>
> Complete circle of fifths
> without seconds: augmented triad, diminished tetrad
> with second: any chromatic chord
>
> I know it is not so scientifically based... but to my ears it works.
>
> lorenzo
>

🔗Lorenzo Frizzera <lorenzo.frizzera@cdmrovereto.it>

> There are at least two distinct concepts of limit (odd and prime),
> but I don't think your description applies perfectly to either one.
> Both a single harmonic series segment, and a full diamond with enough
> pitches to allow 1/1 to serve as every possible harmonic in this
> segment as part of a full harmonic chord in each case, have the same
> limit, according to both concepts, even though the latter has almost
> the square of the number of pitches of the former.

Considering the 5-limit we have this basic harmonic series segment: 1 2 3 . 5
This is the trunk of a tree. From each of these numbers begin a branch obtained multiplying each number for the same row: 2 4 6 . 10; 3 6 9 . 15; 5 10 15 . 25;
Repeating the same on each new number gives a big 5-limit tree.
This show the generating way of the limit while its use (ratios) can be similar to a squirrels (musicians) which jumps up and down on the tree.

Since some numbers stay on different branchs (example 30 is on 15, 10 and 6) maybe the better example would be a bush with a lot of crossed branchs...

>> We can consider consonant an interval only if greater than a minor
> third (I
>> know it is very arguable). This has to do with the distance and the
>> distribution between notes. From this point of view the 7/b9 chord
> is the
>> widest pentad because there are just two intervals of second (one
> major and
>> one minor) while the 9 chord has three of them (major).
>
> But I could argue that no major second dyad has as much critical-band
> roughness as a triad in which all notes fall into a minor third span,
> especially as you are ignoring overtones.

In this case it would be useful to quantify the difference between the roughness of two major seconds and that of a minor second.

Lorenzo

[tuning] Flat ninth

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