Yahoo Tuning Groups Ultimate Backup tuning Dominant b5 justification

A couple years ago I was asking about the b5 note in scale/melody context.
At the time I didn't really grasp the dynamic and flexible function of melody.
I've got a much better sense of things now.

I'm solid on understanding all the harmonic relationships of the basic
harmonic series up to at least 21-odd limit with some understanding beyond
that. I mean understanding in terms of really making sense of the musical
value and having an inate sense of the effect of the different relationships.

I've got a good understanding of the compromises and issues and details
of relating traditional theory to tuning theory and harmonics.

For example, my feeling about the minor chord with a major 7 is clearly
that it is a dissonant, relatively unstable chord but I can see how (such as shown
on a lattice) it has harmonic relationships.

I also understand, such as in Schenkerian theory, how voice-leading can bring
about harmonic structures that really do not harmonically function on their own
so much as are just the result of voice leading between more stable harmonies.

However, after looking through a bunch of jazz theory, there is still ONE chord
that *sounds* like a stable harmony to me, but I have no sense of how it is
harmonically justified: The dominant chord with a b5.

Sure, it has no minor 2nd dissonances, and in ET it is curiously root flexible,
similarly (though not the same) as ET diminished 7 chords. But like dim chords
being related to 17th harmonic, it sounds harmonically related and stable to me.
What is going on? Is there a real harmonic justification for a dominant b5 chord?
It just doesn't seem to be one of those "just due to voice leading" chords...

Thanks everyone.

-Aaron

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

Aaron Wolf escreveu:

> However, after looking through a bunch of jazz theory, there is still ONE chord
> that *sounds* like a stable harmony to me, but I have no sense of how it is
> harmonically justified: The dominant chord with a b5.

It resembles to me (by enharmony) a aug11 chord:

4:5:7:11

> > Sure, it has no minor 2nd dissonances, and in ET it is curiously root flexible,
> similarly (though not the same) as ET diminished 7 chords.

For instance : Eb G A Db (1379) t(1)(4-25)
Is like: A C# D# G (1379) t(1)(4-25)

> But like dim chords
> being related to 17th harmonic, it sounds harmonically related and stable to me.

They can be stable out of any clear tonal context.

> What is going on? Is there a real harmonic justification for a dominant b5 chord?
> It just doesn't seem to be one of those "just due to voice leading" chords...
> > Thanks everyone.
> > -Aaron

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🔗Carl Lumma <clumma@yahoo.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > However, after looking through a bunch of jazz theory, there is
> > still ONE chord that *sounds* like a stable harmony to me, but
> > I have no sense of how it is harmonically justified: The
> > dominant chord with a b5.
>
> What's wrong with 8:10:11:14?
>
> -Carl
>

That's the lowest-term proportion which gives any resemblance
to the "V b5" chord ... but of course the 11 is actually about
halfway between a perfect-4th and a b5.

A much better rational analysis, at least of the 12-edo version
of a b5 chord, is 19:24:27:34.

I'm sure there are lots of maths that could show you this,
but me being a very visual person, i made an Excel spreadsheet
which plots rational approximations of EDO chords on a graph:

/tuning/files/monz/graphic-harmonic-analyzer.xls

You're not limited to 12-edo, you can try and EDO at all.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Carl Lumma <clumma@yahoo.com>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > A much better rational analysis, at least of the 12-edo version
> > of a b5 chord, is 19:24:27:34.
>
> What makes it better? I'm not convinced that intervals
> like 27:19 have a whole lot of meaning... in a chord,
> maybe, especially voiced as 12:17:19:27 or 6:17:19:27.
>
> > I'm sure there are lots of maths that could show you this,
> > but me being a very visual person, i made an Excel spreadsheet
> > which plots rational approximations of EDO chords on a graph:
> >
> > /tuning/files/monz/
> > graphic-harmonic-analyzer.xls
>
> You mean it's closer?

Yes. What i mean is that the proportion 19:24:27:34 is much
closer to the actual proportions of the 12-edo chord.

And since the original post referred to jazz harmony,
i would say that the 12-edo version of Vb5 is probably
what was meant. It's true that jazz horn and guitar players
(and singers) bend pitches all over the place, but as for
harmony, it's pretty much 12-edo piano.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Aaron Wolf <backfromthesilo@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗monz <monz@tonalsoft.com>

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Wolf" <backfromthesilo@> wrote:
> >
> > I came up with a reasonable alternative that I'm not
> > certain about.
> >
> > What do you think of this possibility:
> >
> > 1/1, 5/4, 7/5, 7/4
> >
> > It seems obvious now that I think about it. To me, a
> > primary factor of the sound of the ET jazz chord is that
> > the flat fifth harmonizes as a 3rd with the 7th of the
> > chord. This interpretation seems no further from ET than
> > a 4:5:6:7 chord used for a normal dominant.
>
>
> In fact, it's extremely close to the proportions i posted
> which are a close approximation to the 12-edo chord:
>
> 1/1, 5/4, 7/5, 7/4 = 20:25:28:35 (exactly)
>
> 12-edo Vb5 = ~19:24:27:34

Why talk about 12-edo when you can do it in 6-edo?

My suspicion is that nothing but the general harmonic properties of
the whole-note scale is underlying this and a few other types of
'jazz' chord.

- eg one I found in Noel Coward: G-F-B-D#-A (upwards from the bass)

simply five out of six. Formally the whole-tone scale has two
'dissonances', the tone and the tritone, but they are pretty mild by
modern standards and even more so if retuned to 7-limit. (Though the A
in this case would be the slightly odd 35:32 if we do so and ask for
F-A a pure third -> 8:14:20:25:35

... or 36 if F-A is to be 7:9.)

Now considering other chords made up of four whole-tone pitches, is
G-B-Db-F really more 'stable' or 'consonant' than G-A-C#-F or G-B-D#-F
(respelling each to respect the major thirds)?

~~~T~~~

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>However, after looking through a bunch of jazz theory, there is >still ONE chord
>that *sounds* like a stable harmony to me, but I have no sense of how it is
>harmonically justified: The dominant chord with a b5.
>
>Sure, it has no minor 2nd dissonances, and in ET it is curiously >root flexible,
>similarly (though not the same) as ET diminished 7 chords. But like >dim chords
>being related to 17th harmonic, it sounds harmonically related and >stable to me.
>What is going on? Is there a real harmonic justification for a >dominant b5 chord?
>It just doesn't seem to be one of those "just due to voice leading" chords...

I think you're asking for too much if you want to do either of the following:

1) Derive sensory consonance from simple tuning lattices. It's not at all clear how (or that) lattice distances relate to (context-free) judgments of consonance and dissonance. I agree there might be a connection here, but we're very far from understanding what it is.

2) Derive musical stability from sensory consonance. Different styles have different conceptions of "stability" and these relate in complicated ways to the physical facts. Hence, the fourth was once stable, then became only quasi-stable in the tonal era. The dominant seventh flat five is a very common jazz chord, and that is probably one reason why it sounds stable.

Honestly, this whole discussion about the b5 dominant chord in jazz strikes me as pretty fanciful. Sensory consonance is very, very complex and musical consonance is probably even more so. They're great topics to study -- but you need to do much more than just guess about some ratios, or plot the chord on your favorite tuning lattice.

PS. I've been overall quite impressed with the level of conversation on this list and on tuning-math. I just think here people are getting a bit ahead of themselves.
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

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

Hi Dmitri.

Dmitri Tymoczko escreveu:

> I think you're asking for too much if you want to do either of the following:
> > 1) Derive sensory consonance from simple tuning lattices. It's not > at all clear how (or that) lattice distances relate to (context-free) > judgments of consonance and dissonance. I agree there might be a > connection here, but we're very far from understanding what it is.
> > 2) Derive musical stability from sensory consonance. Different > styles have different conceptions of "stability" and these relate in > complicated ways to the physical facts. Hence, the fourth was once > stable, then became only quasi-stable in the tonal era. The dominant > seventh flat five is a very common jazz chord, and that is probably > one reason why it sounds stable.

I just recall that there is a (somewhat subjective) approach to estimate dissonance (proposed by Krenï¿½k, I think) in which:

8J
5J
4J
are perfect consonances (level 0).
3m
3M
6m
6M
are imperfect consonances (level 1).
2M
7m
are soft dissonances (level 2).
7M
2m
are hard dissonances (level 3).
4A
5dim
are vague intervals:
can be either consonances or dissonances
according to the context.

(Enharmonic intervals are taken as equivalent.)

Therefore:

A chord like {C E Gb Bb} contains:

* No perfect consonance;
* 2 imperfect consonances: {C E} and {Gb Bb};
* 2 soft dissonances: {E Gb} and {C Bb};
* No hard dissonances;
* 2 vague intervals: {C Gb} and {E Bb};

In a context in which can predominate dissonant chords (e.g. using hard dissonances), a chord like {C E G Bb} can indeed sound ``stable'', once it contains only two dissonances, and of the soft kind.

> > Honestly, this whole discussion about the b5 dominant chord in jazz > strikes me as pretty fanciful. Sensory consonance is very, very > complex and musical consonance is probably even more so. They're > great topics to study -- but you need to do much more than just guess > about some ratios, or plot the chord on your favorite tuning lattice.
> > DT
> > PS. I've been overall quite impressed with the level of conversation > on this list and on tuning-math. I just think here people are > getting a bit ahead of themselves.

Wholesome and opportune statement. ;-)

Hudson

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🔗Carl Lumma <clumma@yahoo.com>

> 1) Derive sensory consonance from simple tuning lattices.
> It's not at all clear how (or that) lattice distances relate
> to (context-free) judgments of consonance and dissonance. I
> agree there might be a connection here, but we're very far
> from understanding what it is.

For dyads it's well understood. Without octave equivalence,
distance on the Tenney lattice (which is equivalent to log(n*d)
for a dyad n/d) equates to dissonance for harmonic timbres.
With octave equivalence, distance on the 'Kees' lattice does
the same. Paul Erlich's shown all this, including that log(n*d)
agrees with "harmonic entropy" for JI intervals.

For chords it's a different story. But the usual consonant
triads and tetrads from 12-equal are compact on the 5-limit
lattice.

> 2) Derive musical stability from sensory consonance.

Not possible, in my view.

> Honestly, this whole discussion about the b5 dominant chord
> in jazz strikes me as pretty fanciful. Sensory consonance is
> very, very complex and musical consonance is probably even
> more so. They're great topics to study -- but you need to do
> much more than just guess about some ratios, or plot the chord
> on your favorite tuning lattice.

Again terminology is difficult. Here I think you're using
"sensory consonance" to mean context-free consonance. That's
probably what Plomp, Levelt, Sethares, et al did in their
psychoacoustics. But inevitably the term has become tied to
their particular brand of consonance, based on critical band
collisions. But the critical band can't explain the whole
picture of context-free consonance. Probably more important
is the virtual pitch phenomenon, whereby the brain attempts
to fit the spectra it hears into a harmonic series and label
that spectra with the fundamental of that series. Both
mechanisms are at work in consonance judgements.

Easley Blackwood suggested the terms "concordance" for
context-free consonance, and "consonance" for the perception
in musical context. But this suggestion never really caught
on. So if somebody says "sensory dissonance" we never know
if they mean critical band stuff, or more generally something
outside of a piece of music.

Anyway, my answer to Aaron Wolf's question was a shot at
what the virtual pitch processor might call the chord in
question. Harmonic entropy provides a way to quantify this.
Unfortunately it's a mess to calculate for chords.

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>For dyads it's well understood. Without octave equivalence,
>distance on the Tenney lattice (which is equivalent to log(n*d)
>for a dyad n/d) equates to dissonance for harmonic timbres.
>With octave equivalence, distance on the 'Kees' lattice does
>the same. Paul Erlich's shown all this, including that log(n*d)
>agrees with "harmonic entropy" for JI intervals.

I really think it's much more complicated than your making things out to be. For one thing, dissonance depends on: loudness, absolute pitch height, the specifics of the timbre you're using, and probably many other things besides. Furthermore, what's needed are real empirical studies -- it's not going to be something one "shows" in an a priori manner.

I don't know Erlich's or Tenney's work as well as I would like to, and so can't comment further.

>For chords it's a different story. But the usual consonant
>triads and tetrads from 12-equal are compact on the 5-limit
>lattice.

Agreed. And this is suggestive. But there's a big leap from this simple observation to a deeper understanding of why this might be so. The deeper understanding is precisely what's needed in order to think about the dissonance of non-familiar chords.

>Here I think you're using
>"sensory consonance" to mean context-free consonance. That's
>probably what Plomp, Levelt, Sethares, et al did in their
>psychoacoustics. But inevitably the term has become tied to
>their particular brand of consonance, based on critical band
>collisions. But the critical band can't explain the whole
>picture of context-free consonance. Probably more important
>is the virtual pitch phenomenon, whereby the brain attempts
>to fit the spectra it hears into a harmonic series and label
>that spectra with the fundamental of that series. Both
>mechanisms are at work in consonance judgements.

Yeah, agreed -- though I wouldn't say we even know whether "virtual pitch" is more or less important. There's some real, serious cognitive science/psychology/neuroscience to be done here. A really good quantitative model of dissonance, along with a plausible story about its neurological mechanisms, would be a legitimate candidate for a Nobel prize.

Until that serious science is done, theorists (academic or non-academic) can make up all sorts of models, make all sorts of declarations, and generally pronounce as they fit about consonance and dissonance. But in my view all of this is really just going to be guesswork, and I doubt it's worth taking very seriously.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Easley Blackwood suggested the terms "concordance" for
> context-free consonance, and "consonance" for the perception
> in musical context. But this suggestion never really caught
> on. So if somebody says "sensory dissonance" we never know
> if they mean critical band stuff, or more generally something
> outside of a piece of music.

I never knew that it was Blackwood who suggested that!
(That's strange, over the past few weeks i've been re-engaging
with his work again ...)

As can be seen from my Encyclopedia (when i get it back online
again), i am a staunch proponent of adhering to this convention:

"Accordance", a continuum of affect whose opposite poles
are concordance and discordance, refers to a psychoacoustical
context-free perception;

"Sonance", a continuum of affect whose opposite poles
are consonance and dissonance, refers to a perception
based on musical context.

Note that the opposite poles are at ends of a continuum, and
are not considered to be two different kinds of perception.
I.e., they are differences in degree and not in kind.
Many other music-theory text speak of them as the latter.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Aaron Wolf <backfromthesilo@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > Easley Blackwood suggested the terms "concordance" for
> > context-free consonance, and "consonance" for the perception
> > in musical context. But this suggestion never really caught
> > on. So if somebody says "sensory dissonance" we never know
> > if they mean critical band stuff, or more generally something
> > outside of a piece of music.
>
>
> I never knew that it was Blackwood who suggested that!
> (That's strange, over the past few weeks i've been re-engaging
> with his work again ...)
>
> As can be seen from my Encyclopedia (when i get it back online
> again), i am a staunch proponent of adhering to this convention:
>
> "Accordance", a continuum of affect whose opposite poles
> are concordance and discordance, refers to a psychoacoustical
> context-free perception;
>
> "Sonance", a continuum of affect whose opposite poles
> are consonance and dissonance, refers to a perception
> based on musical context.
>
> Note that the opposite poles are at ends of a continuum, and
> are not considered to be two different kinds of perception.
> I.e., they are differences in degree and not in kind.
> Many other music-theory text speak of them as the latter.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

Just a quick note that I think this is wonderful. Monz, I applaud you for
working in a pragmatic way to improve the language we use to understand
and teach these basic elements of music.

-Aaron

🔗Aaron Wolf <backfromthesilo@yahoo.com>

A general response to all the great discussion:

As a teacher and theorist, I'm extremely interested in understanding
the continuum between universal constants and completely arbitrary
conventions.

While there will always be unanswered questions, my goal is to be able
to explain the factors that would place a musical item on that continuum.

For example, the major chord is certainly near the universal end, and this
can be explained by the simple nature of harmonic timbres particularly
the human voice. But we can still discuss how its prominance and particular
uses may be circumstantial conventions in some respects.

The various factors that govern perception of fusion or fission of elements
in the mind of a listener are universal factors. How they apply to particular
circumstances is a complex question that we may not always be able to
answer.

In the case of #9 chord, I can cite the 19th harmonic, also discuss leading tone
and voice leading issues, and then still leave the question open as to what is
really being perceived. Clearly there are *some* universal or near universal
factors that favor the existence of the chord.

On the other hand, the augmented chord seems to me to be clearly the result
of two factors: voice leading, and dumb theorists saying "oh, if we have a major
third and a minor third in a major chord, what if we just have two major thirds?"

In context, any sound or combination of sound can have meaning in how it
relates to the music around it. My question is whether some chords or
progressions (or rhythms for that matter) have prominance due to factors
beyond convention.

My feeling about many jazz chords is that they came to be simply by
musicians arbitrarily adding every other note, or doing other such experimental
randomness and then finding a musical use for it. Or that they came to
be simply as a result of voice-leading. Hence the idea of 11th and 13th chord
names, which are namewise just ridiculous BS even if someone found a way to
make the chords have a musical place.
Clearly, that ie not the case with open 5ths or with dominant chords etc. etc.

The dominant b5 sounds to me like it is easier to come upon than the more
ridiculous jazz chords. That it sounds supported enough to possibly be something
a bunch of singers could come upon just making things up by ear. Whereas
many chords only exist by intellectual games of amateur theorists trying things.
I wonder if others agree or if I'm kidding myself. And my feeling is that if it really
could be an identifiable relationship to musicians who play by ear and know nothing
of theory who did NOT learn it by convention (again I don't know if this could be)
then it must be because there is *some* universal factor justifying it.

-Aaron

P.S. I know I'm a little hipocritical trying to come up with explanations for
everything while bashing people who's simplistic thinking got us "13th" chords in
ET. Well, unless I'm sure an explanation really makes sense, I won't go around
tauting it, so at least I can temper myself...

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Aaron Wolf" <backfromthesilo@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > As can be seen from my Encyclopedia (when i get it back online
> > again), i am a staunch proponent of adhering to this convention:
> >
> > "Accordance", a continuum of affect whose opposite poles
> > are concordance and discordance, refers to a psychoacoustical
> > context-free perception;
> >
> > "Sonance", a continuum of affect whose opposite poles
> > are consonance and dissonance, refers to a perception
> > based on musical context.
> >
> > Note that the opposite poles are at ends of a continuum, and
> > are not considered to be two different kinds of perception.
> > I.e., they are differences in degree and not in kind.
> > Many other music-theory text speak of them as the latter.
> >
> >
>
> Just a quick note that I think this is wonderful. Monz,
> I applaud you for working in a pragmatic way to improve
> the language we use to understand and teach these basic
> elements of music.
>
> -Aaron

Thanks, Aaron, i appreciate that.

Regarding my last paragraph, i thought it worth pointing out
three other notable music-theorists before me who treated
consonance/dissonance as a continuum of affect:
Helmholtz, Schoenberg, and Partch.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Carl Lumma <clumma@yahoo.com>

> > As can be seen from my Encyclopedia (when i get it back online
> > again), i am a staunch proponent of adhering to this convention:
> >
> > "Accordance", a continuum of affect whose opposite poles
> > are concordance and discordance, refers to a psychoacoustical
> > context-free perception;
> >
> > "Sonance", a continuum of affect whose opposite poles
> > are consonance and dissonance, refers to a perception
> > based on musical context.
> >
> > Note that the opposite poles are at ends of a continuum, and
> > are not considered to be two different kinds of perception.
> > I.e., they are differences in degree and not in kind.
> > Many other music-theory text speak of them as the latter.

> Just a quick note that I think this is wonderful. Monz, I
> applaud you for working in a pragmatic way to improve the
> language we use to understand and teach these basic elements
> of music.

I also applaud it. However, it seems more suited to an
expository article than an encyclopedia.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

> >For dyads it's well understood. Without octave equivalence,
> >distance on the Tenney lattice (which is equivalent to log(n*d)
> >for a dyad n/d) equates to dissonance for harmonic timbres.
> >With octave equivalence, distance on the 'Kees' lattice does
> >the same. Paul Erlich's shown all this, including that log(n*d)
> >agrees with "harmonic entropy" for JI intervals.
>
> I really think it's much more complicated than your making
> things out to be. For one thing, dissonance depends on:
> loudness, absolute pitch height, the specifics of the timbre
> you're using, and probably many other things besides.

All things you can plug into psychoacoustic models of critical
band collisions -- combination tones, too. Not harmonic
entropy though (that I know of). But harmonic entropy is a
very high-level concept that presumably models very high-level
signal processing in the brain (there's even a hand-waving
neuroanatomy explanation, though my feeling is Paul's been
avoiding it lately because he doesn't think it's needed).

All of what I said is qualified for harmonic timbres only. If
you can swallow octave-equivalence, ignoring the other things
you mention is no problem. And anyway it isn't much of a problem,
in the range of amplitudes and timbres commonly seen in the
practice of western music. All of this stuff is meant to
describe relative changes in a given instrumentation -- not to
measure some quantity to compare different instrumentations.

Let's not forget spatial cues. But mixing stereo to mono
isn't a disaster (even though that practice is much harder on
the signal than just using a single mic, due to phase
cancelations).

Think about how much most speakers color sound. The harmonic
distortion of a tube amp. Or buy an "effects" unit or synth
and run some recordings through resonant filters. You have
to go way outside of normal musical bounds to get something
that sounds different. Which is what people who use sythesizers
tend to do.

Or sit at a mixing board and change the relative levels of the
instruments (a growing number of orchestral recordings are
multitracked these days). It doesn't change the fundamentals
of consonance for the music.

> I don't know Erlich's or Tenney's work as well as I would like
> to, and so can't comment further.

Try

http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937

> >For chords it's a different story. But the usual consonant
> >triads and tetrads from 12-equal are compact on the 5-limit
> >lattice.
>
> Agreed. And this is suggestive. But there's a big leap from this
> simple observation to a deeper understanding of why this might be
> so.

From a critical-band-collision POV, the reason is the component
dyads are all JI. From a harmonic entropy point of view, well,
we don't really know, as Gene points out. But it favors some
compact lattice structures over others... so it does make
predictions. In particular, it tends to favor harmonic chords
over subharmonic ones (their inversions as you would say). This
isn't very evident in the 5-limit, but it's already very obvious
in the 7-limit (try it and JI and you'll see).

> >Here I think you're using
> >"sensory consonance" to mean context-free consonance. That's
> >probably what Plomp, Levelt, Sethares, et al did in their
> >psychoacoustics. But inevitably the term has become tied to
> >their particular brand of consonance, based on critical band
> >collisions. But the critical band can't explain the whole
> >picture of context-free consonance. Probably more important
> >is the virtual pitch phenomenon, whereby the brain attempts
> >to fit the spectra it hears into a harmonic series and label
> >that spectra with the fundamental of that series. Both
> >mechanisms are at work in consonance judgements.
>
> Yeah, agreed -- though I wouldn't say we even know whether
> "virtual pitch" is more or less important. There's some real,
> serious cognitive science/psychology/neuroscience to be done
> here. A really good quantitative model of dissonance, along
> with a plausible story about its neurological mechanisms,
> would be a legitimate candidate for a Nobel prize.

Then Martin Braun will maybe win one day. Too bad there
isn't a prize for neuroscience.

> But in my view all of this is really just going to
> be guesswork, and I doubt it's worth taking very seriously.

When one starts hearing and composing in alternate tunings,
one sometimes finds it isn't guesswork. Unconstrained by a
system where there's only so much dissonance available, one
may find out pretty quickly. There are mysteries, but there
are also very practical abstractions that are immediate.

There was also an impromptu blind listening test on the
harmonic entropy list.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

🔗yahya_melb <yahya@melbpc.org.au>

Hey Monz,

--- In tuning@yahoogroups.com, "monz" wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" wrote:
> >
> > > However, after looking through a bunch of jazz theory, there is
> > > still ONE chord that *sounds* like a stable harmony to me, but
> > > I have no sense of how it is harmonically justified: The
> > > dominant chord with a b5.
> >
> > What's wrong with 8:10:11:14?
>
> That's the lowest-term proportion which gives any resemblance
> to the "V b5" chord ... but of course the 11 is actually about
> halfway between a perfect-4th and a b5.
>
> A much better rational analysis, at least of the 12-edo version
> of a b5 chord, is 19:24:27:34.
>
> I'm sure there are lots of maths that could show you this,
> but me being a very visual person, i made an Excel spreadsheet
> which plots rational approximations of EDO chords on a graph:
/tuning/files/monz/graphic
-harmonic-analyzer.xls
>
> You're not limited to 12-edo, you can try and EDO at all.

Hmmm, I'm pretty visual, too, so I looked at your worksheet, then
punched in 0 5 7 10, following instructions, which say:
"input cardinality of EDO, then degree numbers of notes in chord
spreadsheet calculates harmonic analysis"

What the sheet does is to calculate a column of values for each
input number of degrees (eg 9 above), then plot the results on a
graph. But I'm lost as to what to do with all this - I can't answer
the questions: What is the point of the graph? & how should one read
it to extract useful musical information?

Regards,
Yahya

🔗yahya_melb <yahya@melbpc.org.au>

Hi Aaron,

--- In tuning@yahoogroups.com, "Aaron Wolf" wrote:
>
> A general response to all the great discussion:
>
> As a teacher and theorist, I'm extremely interested in
understanding the continuum between universal constants and
completely arbitrary conventions.

*That* sentence took some digesting! But .. are you sure there's a
continuum? I'd have thought "a disconnect" would be more likely.

> While there will always be unanswered questions, my goal is to be
able to explain the factors that would place a musical item on that
continuum.
>
> For example, the major chord is certainly near the universal end,
and this can be explained by the simple nature of harmonic timbres
particularly the human voice. But we can still discuss how its
prominance and particular uses may be circumstantial conventions in
some respects.

IMO this is true, no matter how you choose to connect the two sets
of facts.

> The various factors that govern perception of fusion or fission of
elements in the mind of a listener are universal factors. ...

Really? What leads you that conclusion?

> ... How they apply to particular circumstances is a complex
question that we may not always be able to answer.

Rarely will be able, IMO.

> In the case of #9 chord, I can cite the 19th harmonic, also
discuss leading tone and voice leading issues, and then still leave
the question open as to what is really being perceived. Clearly
there are *some* universal or near universal factors that favor the
existence of the chord.

Again, query universality.

> On the other hand, the augmented chord seems to me to be clearly
the result of two factors: voice leading, and dumb theorists
saying "oh, if we have a major third and a minor third in a major
chord, what if we just have two major thirds?"

And the second seems to me quite likely; even likelier, is dumb NON-
theorists just playing around on the keyboard or fretboard and
coming up with new chords that "work" in progressions.

> In context, any sound or combination of sound can have meaning in
how it relates to the music around it. My question is whether some
chords or progressions (or rhythms for that matter) have prominance
due to factors beyond convention.

Your assumption of universals surely is an assumption that they do?

> My feeling about many jazz chords is that they came to be simply
by musicians arbitrarily adding every other note, or doing other
such experimental randomness and then finding a musical use for it...

Yep. that's more or less what I said above. I've often found some
of the sounds I use by simple experiment, turning off the theorising
brain and opening the ear. You know, just sitting in front of the
keyboard, or picking up a guitar or flute, _closing the eyes_ and
flinging the fingers around with wild abandon! When you hear
something you like, do it again. If you still like it, take a peek
and find out *what* you were doing, so you can do it again later.

> ... ridiculous BS ... the more ridiculous jazz chords ... many
chords only exist by intellectual games of amateur theorists trying
things.

I don't know that the pejorative tone is useful in a serious
discussion of musical origins. Many people have, over the
millennia, done things just because they liked the sound of them;
others have done them based on wacky and since-discredited
theories. Does that make them any less valuable as musical impulses?

> I wonder if others agree or if I'm kidding myself. And my feeling
is that if it really could be an identifiable relationship to
musicians who play by ear and know nothing of theory who did NOT
learn it by convention (again I don't know if this could be) then it
must be because there is *some* universal factor justifying it.

Aaron, the only "universals" I can come up with for you are that
(a) people have always expressed themselves in music and
(b) people like to play.

There is also some evidence that play is a useful learning strategy
for many animals, humans included.

As a consequence of such play, I for one rediscovered many chords
for myself (including major, minor, diminished and augmented and
others I still don't know the "proper" names for) at a fairly young
age, and in a total theoretical vacuum.

> P.S. I know I'm a little hipocritical trying to come up with
explanations for everything while bashing people who's simplistic
thinking got us "13th" chords in ET. Well, unless I'm sure an
explanation really makes sense, I won't go around tauting it, so at
least I can temper myself...

"The Well-Tempered Wolf"! ;-)

Regards,
Yahya

🔗monz <monz@tonalsoft.com>

Hi Yahya,

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
>
> > <snip>
> > but me being a very visual person, i made an Excel spreadsheet
> > which plots rational approximations of EDO chords on a graph:
>
/tuning/files/monz/graphic-harmonic-analyzer.xls
> >
>
> <snip>
>
> What the sheet does is to calculate a column of values for
> each input number of degrees (eg 9 above), then plot the
> results on a graph. But I'm lost as to what to do with all
> this - I can't answer the questions: What is the point of
> the graph? & how should one read it to extract useful musical
> information?

The graph plots the proportion values of each chord vertically.

You didn't say what cardinality you chose, only that you entered
the values 0 5 7 10 for the degrees of the chord members. So i
kept the cardinality at 12 (edo) and took a look ...

If you look at the graph, the column which has 6.0 as the bottom
note shows that the bottom three notes of this chord are almost
exactly a 6:8:9 proportion. The graph shows the 10th degree as
a ratio of 6.0:10.7, so the closest integer proportion using
this interpretation is 6:8:9:11, but the 11 is rather far off.

Looking for a better fit, the one that shows the least error
from integer proportions is the column whose lowest note = 18.0.
This give an approximate integer proportion of 18:24:27:32 for
the chord.

Just ask if you have more questions.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Wolf" <backfromthesilo@>
wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> > >
> > > As can be seen from my Encyclopedia (when i get it back online
> > > again), i am a staunch proponent of adhering to this convention:
> > >
> > > "Accordance", a continuum of affect whose opposite poles
> > > are concordance and discordance, refers to a psychoacoustical
> > > context-free perception;
> > >
> > > "Sonance", a continuum of affect whose opposite poles
> > > are consonance and dissonance, refers to a perception
> > > based on musical context.
> > >
> > > Note that the opposite poles are at ends of a continuum, and
> > > are not considered to be two different kinds of perception.
> > > I.e., they are differences in degree and not in kind.
> > > Many other music-theory text speak of them as the latter.
> >
> > Just a quick note that I think this is wonderful. Monz,
> > I applaud you for working in a pragmatic way to improve
> > the language we use to understand and teach these basic
> > elements of music.
> >
> > -Aaron
>
> Thanks, Aaron, i appreciate that.
>
> Regarding my last paragraph, i thought it worth pointing out
> three other notable music-theorists before me who treated
> consonance/dissonance as a continuum of affect:
> Helmholtz, Schoenberg, and Partch.
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

You may add Paul Hindemith to that list.

--George

🔗yahya_melb <yahya@melbpc.org.au>

Hi monz,

--- In tuning@yahoogroups.com, "monz" wrote:
<snip>
/tuning/files/monz/graphic
-harmonic-analyzer.xls
> > <snip>
> > What the sheet does is to calculate a column of values for
> > each input number of degrees (eg 9 above), then plot the
> > results on a graph. But I'm lost as to what to do with all
> > this - I can't answer the questions: What is the point of
> > the graph? & how should one read it to extract useful musical
> > information?
>
> The graph plots the proportion values of each chord vertically.
>
> You didn't say what cardinality you chose, only that you entered
> the values 0 5 7 10 for the degrees of the chord members. So i
> kept the cardinality at 12 (edo) and took a look ...
>
> If you look at the graph, the column which has 6.0 as the bottom
> note shows that the bottom three notes of this chord are almost
> exactly a 6:8:9 proportion. The graph shows the 10th degree as
> a ratio of 6.0:10.7, so the closest integer proportion using
> this interpretation is 6:8:9:11, but the 11 is rather far off.
>
> Looking for a better fit, the one that shows the least error
> from integer proportions is the column whose lowest note = 18.0.
> This give an approximate integer proportion of 18:24:27:32 for
> the chord.
>
> Just ask if you have more questions.

Thanks, that's much clearer. I guess I could give the cells a
colouring contingent on their [relative] error from the nearest
integer. Or, turning it around, for any user-adjustable tolerance,
have the worksheet produce the smallest integer proportions that lie
within that tolerance (no graph necessary). For example, given the
input 12; 0 5 7 10; 0.10 % (say), we might have the worksheet show
the chord ratios as 18:24:27:32.

Regards,
Yahya

🔗monz <monz@tonalsoft.com>

Hi Yahya,

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
>
> Hi monz,

/tuning/files/monz/graphic-harmonic-analyzer.xls

> Thanks, that's much clearer. I guess I could give the
> cells a colouring contingent on their [relative] error
> from the nearest integer. Or, turning it around, for any
> user-adjustable tolerance, have the worksheet produce
> the smallest integer proportions that lie within that
> tolerance (no graph necessary). For example, given the
> input 12; 0 5 7 10; 0.10 % (say), we might have the
> worksheet show the chord ratios as 18:24:27:32.

Certainly, one could easily harness the power of Excel
to perform calculations which do all the work for you.

But as i said, i'm a visually-oriented person, and i
like looking at the graphs myself to make the determinations.

-monz
http://tonalsoft.com
Tonescape microtonal music software