dissonance/consonance

> Right now I have to finish preparing the slide show for tomorrow's
> class session on, as fate would have it, the acoustic basis for
> consonance and dissonance.

Is that the "critical band" stuff by Plomp et al, with sine waves, or are you bringing in additional context such as style, function, convention, and timbre?

Try this question that I've sent to two university music theory instructors for comment:

The context is two-voiced texture (melody + bass), typical tonal music c1600-c1850. Disregard any and all issues of keyboard temperament.
Are augmented seconds, augmented fifths, and diminished fourths ALWAYS dissonances, while minor thirds, minor sixths, and major thirds are always consonances? So, on keyboard, we've got exactly the same pair of pitches being played.......

Brad Lehman

Brad Lehman wrote:
> > > > Right now I have to finish preparing the slide show for tomorrow's
> > class session on, as fate would have it, the acoustic basis for
> > consonance and dissonance.
> > Is that the "critical band" stuff by Plomp et al, with sine waves, or
> are you bringing in additional context such as style, function,
> convention, and timbre?
> > Try this question that I've sent to two university music theory
> instructors for comment:
> > The context is two-voiced texture (melody + bass), typical tonal music
> c1600-c1850. Disregard any and all issues of keyboard temperament.
> Are augmented seconds, augmented fifths, and diminished fourths ALWAYS
> dissonances, while minor thirds, minor sixths, and major thirds are
> always consonances? So, on keyboard, we've got exactly the same pair of
> pitches being played.......
> > Brad Lehman
> > William Sethares wrote a whole book on the subject, "Tuning, Timbre, Spectrum, Scale". It's in the second edition.

The truth of the matter is that there is not a clear answer. More good might be done by framing it as such. Perhaps someone there will get closer. The models we have, many get close but always seem to fall short. John Chalmers' The Division of the tetrachord list quite a few 'systems' in analyzing tetrachords which correctly focuses on how elusive it is. Helmholtz analyzes things according to roughness and is a good starting point cause this aspect of his work is ignored. It takes into consideration range for one and how an interval changes in its range. The implication is that a 6/5 above C1 is not the same as one above G1. This is just with the same timbre. Sethares work points out how the subject gets evens more complex with timbre. For instance how do we rate a sine wave 6/5 against a complex timbre that is adjusted to fit a minor third of some kind that fits. Ernst Toch pointed out also the influence of context how a F# major chord will be not be consonant against C major. Somethings fail to submit to being converted to numerical data even if they point in the right direction. In the end the only judge we have is our ear and taste.

M. Edward (Ed) Borasky wrote:
>
> Brad Lehman wrote:
> >
> >
> > > Right now I have to finish preparing the slide show for tomorrow's
> > > class session on, as fate would have it, the acoustic basis for
> > > consonance and dissonance.
> >
> > Is that the "critical band" stuff by Plomp et al, with sine waves, or
> > are you bringing in additional context such as style, function,
> > convention, and timbre?
> >
> > Try this question that I've sent to two university music theory
> > instructors for comment:
> >
> > The context is two-voiced texture (melody + bass), typical tonal music
> > c1600-c1850. Disregard any and all issues of keyboard temperament.
> > Are augmented seconds, augmented fifths, and diminished fourths ALWAYS
> > dissonances, while minor thirds, minor sixths, and major thirds are
> > always consonances? So, on keyboard, we've got exactly the same pair of
> > pitches being played.......
> >
> > Brad Lehman
> >
> >
>
> William Sethares wrote a whole book on the subject, "Tuning, Timbre,
> Spectrum, Scale". It's in the second edition.
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Brad Lehman <bpl@...> wrote:
>
> > Right now I have to finish preparing the slide show for tomorrow's
> > class session on, as fate would have it, the acoustic basis for
> > consonance and dissonance.
>
> Is that the "critical band" stuff by Plomp et al, with sine waves,

The critical band stuff is the essence of it, since it is the c.b.
functions between close pair of overtones which create Helmoholtz's
"roughness"

> or
> are you bringing in additional context such as style, function,
> convention,

no, no, and no. The topic was the ACOUSTICAL basis for consonance and
dissonance. We did have a good long talk about how every musical
culture indoctrinates musicians and listeners to stop listening to the
pure acoustical qualities in one way or anther and replace the raw
perception with the often artificial trimmings and trappings of a
musical syntax... unless, of course, you are some maverick like Cage
or Partch who manage to get beyond it all and start hearing again.

> and timbre?

of course! Timbre is the key! Without timbre, there is nothing to
discuss, as everything wider than the c.b. would be equally consonant.

One of the things I have the kids do is listen to the sound of a pure
major third played first by two tones including all harmonics, then by
one tone with only odd harmonics, first in the lower position and then
in the upper position, and finally two tones with only odd harmonics.
They are always amazed at how the quality of the interval changes so much

It's rather like studying orchestration, actually. In that case, you
are not interested in the musical context in which a composer creates
timbres, just in the qualities of the timbres themselves. Ravel's
marvelous borrowing of organ mutation stop technique in Bolero creates
an amazing sound, but the context of the repeated melody over the
static background has nothing whatsoever to do with it.

The Sethares book is a wonderful resource, especially his little
pieces he used to demo how different timbres fit different spectra.
The only place where he is pushing it too far is the whole temperament
chapter, which should have been cut from the book.
>
> Try this question that I've sent to two university music theory
> instructors for comment:
>
> The context is two-voiced texture (melody + bass), typical tonal music
> c1600-c1850. Disregard any and all issues of keyboard temperament.
> Are augmented seconds, augmented fifths, and diminished fourths ALWAYS
> dissonances, while minor thirds, minor sixths, and major thirds are
> always consonances?

Are you speaking acoustically or in terms of functionally? The two are
very different. Also, without knowing what the intonation situation
is, the question is pointless.

Ernst Toch wrote an interesting little book called "The Shaping Forces
in Music", in which he makes the statement that the day will come when
we realize that thinking that certain intervals have certain special
qualities the make them more or less "consonant" or "dissonant" than
other intervals outside of musical context is as stupid and primitive
as believing that certain women are witches and should be burned. He
then proceeds to demonstrate how an octave can function as a point of
unresolved tension. Of course, being an atonalist, he had long since
stopped actually LISTENING to symphonia (a term I prefer over harmony,
with all it's constrained implications), and was completely in the
realm of function. When I was much younger i was convinced by his
little demo; now I can't buy it anymore because I have learned to
HEAR. To much non-western music listening, I fear...

Obviously, Brad, the answer to your question is that acoustically, on
a keyboard instrument, if we assume some sort of neutral temperament
(which is the only thing we CAN assume if temperament is to be kept
out of the issue) there is no difference between an augmented 2nd and
a minor third or a diminished fourth and a major third, so calling the
one or the other consonant or dissonant is utterly pointless. Get your
head out of the score and LISTEN. There is only the acoustic reality.
In terms of musical function, however, there can be a BIG difference.
So what? The answer is so bluhdy obvious I don't understand the point
of posing the question in the first place. Is it some sort of trick to
get university professors pointlessly spinning their intellectual
wheels? Although I would hope they wouldn't be so dumb as to fall for
the trap... rather along the lines of the old no-win question of "Have
you stopped beating your wife yet?"

Ciao,

P

Paul Poletti wrote:

> We did have a good long talk about how every musical
> culture indoctrinates musicians and listeners to stop listening to the
> pure acoustical qualities in one way or anther and replace the raw
> perception with the often artificial trimmings and trappings of a
> musical syntax... unless, of course, you are some maverick like Cage
> or Partch who manage to get beyond it all and start hearing again.

Well ... I guess this is all fodder for debate, but IMHO music composers, performers, and audiences co-evolve in the same way that species co-evolve in ecological niches. Hence, gamelan, Shostakovich, Partch, Cage, Eric Clapton, etc.

> > The Sethares book is a wonderful resource, especially his little
> pieces he used to demo how different timbres fit different spectra.
> The only place where he is pushing it too far is the whole temperament
> chapter, which should have been cut from the book.

I'm not sure I agree -- there's nothing I personally would cut from Sethares. Speaking of Sethares, on the off chance you're not aware of it, he has written another book called _Rhythm and Transforms_. It's mostly about digital signal processing algorithms for "beat tracking", with plenty of audio examples. A fair number of them are based on Maple Leaf Rag.

I think all of his code is in Matlab, which I don't own. It might run in one of the "Matlab clones" like SciLab, Octave or FreeMat. I'll probably just translate the algorithms directly into CSound -- many of them are already built in.

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, Brad Lehman <bpl@> wrote:
> >
> > > Right now I have to finish preparing the slide show
> > > for tomorrow's class session on, as fate would have it,
> > > the acoustic basis for
> > > consonance and dissonance.
> >
> > Is that the "critical band" stuff by Plomp et al, with
> > sine waves,
>
> The critical band stuff is the essence of it, since it
> is the c.b. functions between close pair of overtones
> which create Helmoholtz's "roughness"
>
> > or are you bringing in additional context such as
> > style, function, convention,
>
> no, no, and no. The topic was the ACOUSTICAL basis for
> consonance and dissonance. We did have a good long talk
> about how every musical culture indoctrinates musicians
> and listeners to stop listening to the pure acoustical
> qualities in one way or anther and replace the raw
> perception with the often artificial trimmings and
> trappings of a musical syntax... unless, of course,
> you are some maverick like Cage or Partch who manage
> to get beyond it all and start hearing again.

And i for one, and a few others on this list too, have
been trying for *years* to encourage people to adopt
new terminology so as to avoid this kind of confusion.

Following Blackwood, i've been using "accordance" to
describe the relative concordance / dissonance of an
interval, which is a psychoacoustical perception devoid
of any necessity for considering musical context.

For the latter, i use "sonance" to describe the relative
consonance / dissonance of any number of sonic or musical
objects, be they intervals, chords, cadences,
entire sections of pieces, etc. And this one is entirely
dependent upon consideration of musical context.

It would be so nice if everyone could see the logic
in this and *stick to this terminology*!

I thought that all six of these terms were in my
Encyclopedia, but upon looking now, i see that there
are empty place-holder pages for "discordance" and
"dissonance". Anyway, the other four are there:

http://tonalsoft.com/enc/a/accordance.aspx
http://tonalsoft.com/enc/c/concord.aspx
http://tonalsoft.com/enc/s/sonance.aspx
http://tonalsoft.com/enc/c/consonance.aspx

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

I don't understand this supposed 'context'. Are there any pieces at
all that satisfy the criteria Brad has demanded : exactly two voices
and no influence from keyboard tuning? (Not to mention the fact that
up til 1700 or so quite often there *were* separate keys for aug2 vs.
m3, dim4 vs. M3 etc.)

There are a few Mozart pieces for two horns or violin & viola, but I
strongly doubt these have any of the 'dissonant' intervals he is
claiming to be interested in.

Anyway, independently of whether any pieces exist for which Brad's
question may be asked, he hasn't said what the definition of
'dissonance' is that he wants the poor university profs to stick to.

Marpurg, of all people, has a quite logical approach to what Brad may
be trying to drive at. He defined theoretically dissonant intervals
such as aug2 that on a (nearly-)equal-tempered keyboard are
acoustically identical to theoretically consonant ones such as m3 as
'altered dissonances' (alterierte Dissonanzen).
~~~T~~~

> --- In tuning@yahoogroups.com, Brad Lehman <bpl@> wrote:
> >
> > Try this question that I've sent to two university music theory
> > instructors for comment:
> >
> > The context is two-voiced texture (melody + bass), typical tonal
music
> > c1600-c1850. Disregard any and all issues of keyboard temperament.
> > Are augmented seconds, augmented fifths, and diminished fourths
ALWAYS
> > dissonances, while minor thirds, minor sixths, and major thirds are
> > always consonances?
>
> Are you speaking acoustically or in terms of functionally? The two are
> very different. Also, without knowing what the intonation situation
> is, the question is pointless.

It's critical to distinguish between tonal dissonance and sensory dissonance. Tonal dissonances are dissonant in terms of the musical practice -- within the context of counterpoint and harmony -- while sensory dissonance is a perceptual category. While the one may belong more to a musical domain and the other to a physical domain, there is a remarkable and useful overlap between these two descriptions, with the distinction allowing one to get a handle on matters like being out of tune. For example, when singers or players execute an interval that is recognizeably as both a perfect fifth and an out-of-tune fifth, we are describing both a tonal and a sensory quality.

So, when Brad asks: "Are augmented seconds, augmented fifths, and diminished fourths ALWAYS dissonances, while minor thirds, minor sixths, and major thirds are always consonances? So, on keyboard, we've got exactly the same pair of pitches being played.......", he is getting to the core of the phenomena that in real, existing music, and in particular on instruments of fixed pitch, intervals with greater sensory dissonance can function in the place of intervals with lesser sensory dissonance and vice versa, depending upon the tonal context and our tolerance for interval variation (i.e. how far can one detune an interval from some ideal value for an interval class before one assigns the interval to another class) allows for this. The answer to Brad's question is that, in "well-composed" music of the style in question, augmented seconds, augmented fifths, and diminished fourths will always be tonal dissonances, even when they are acoustically identical -- of equal sensory consonance -- to the minor thirds, minor sixths, and major thirds of another tonal context.

Daniel Wolf
Frankfurt

There are tunings in which the intervals beat at close to the same degree of beating . In this case the differences between say seconds and thirds nd fourths are barely noticeable. The scales derived from recurrent sequences (Mt. Meru) have this as a common feature.

Daniel Wolf wrote:
> It's critical to distinguish between tonal dissonance and sensory > dissonance. Tonal dissonances are dissonant in terms of the musical > practice -- within the context of counterpoint and harmony -- while > sensory dissonance is a perceptual category. While the one may belong > more to a musical domain and the other to a physical domain, there is a > remarkable and useful overlap between these two descriptions, with the > distinction allowing one to get a handle on matters like being out of > tune. For example, when singers or players execute an interval that is > recognizeably as both a perfect fifth and an out-of-tune fifth, we are > describing both a tonal and a sensory quality.
>
> So, when Brad asks: "Are augmented seconds, augmented fifths, and > diminished fourths ALWAYS dissonances, while minor thirds, minor sixths, > and major thirds are always consonances? So, on keyboard, we've got > exactly the same pair of pitches being played.......", he is getting to > the core of the phenomena that in real, existing music, and in particular > on instruments of fixed pitch, intervals with greater sensory dissonance > can function in the place of intervals with lesser sensory dissonance and > vice versa, depending upon the tonal context and our tolerance for > interval variation (i.e. how far can one detune an interval from some > ideal value for an interval class before one assigns the interval to > another class) allows for this. The answer to Brad's question is that, in > "well-composed" music of the style in question, augmented seconds, > augmented fifths, and diminished fourths will always be tonal dissonances, > even when they are acoustically identical -- of equal sensory consonance > -- to the minor thirds, minor sixths, and major thirds of another tonal > context.
>
> Daniel Wolf
> Frankfurt
>
>
>
>
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> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "Daniel Wolf" <djwolf@...> wrote:
>
> It's critical to distinguish between tonal dissonance and sensory
> dissonance. Tonal dissonances are dissonant in terms of the musical
> practice -- within the context of counterpoint and harmony -- while
> sensory dissonance is a perceptual category. While the one may belong
> more to a musical domain and the other to a physical domain, there
is a
> remarkable and useful overlap between these two descriptions
(...)

> in "well-composed" music of the style in question, augmented seconds,
> augmented fifths, and diminished fourths will always be tonal
dissonances,
> even when they are acoustically identical -- of equal sensory
consonance
> -- to the minor thirds, minor sixths, and major thirds of another
tonal
> context.
>
> Daniel Wolf
> Frankfurt

That much is clear, but I still don't understand the question. Brad
seems to want to take some advantage of the acoustical identity on the
keyboard (at least the 12-note keyboard, which was by no means
universal!) while on the other hand asking us to consider two-voiced
compositions independently of any keyboard tuning.

Two-voiced compositions, by definition, do not include a full harmony
- until we have some sort of filling-in parts. If there is no
complete harmony, the musical function of a tempered interval can
often be ambiguous, relying on a wider harmonic context for a tonal
'explanation'. The A minor Prelude of Bach's 'Book 2' (24 Preludes and
Fugues) is a possible example.

Incidentally there are some 'classical-era' compositions for
fixed-pitch instruments which are *not* "well-composed" in Daniel's
sense: notably lute pieces by Silvius Leopold Weiss with enharmonic
modulation via the augmented triad.

The celebrated example of an acoustically consonant functional
dissonance is the perfect fourth above the bass. (Other voicings of
the fourth may be consonant...) No matter how perfectly in tune it is,
it always required a resolution. So of course there is a distinction
between functional dissonance in tonal style and acoustic dissonance.
However, what this should have to do with tuning is not clear.

Historically, in every source I know of in the 'classical' period, the
aim of tuning is to make functional consonances sufficiently
acoustically consonant. Functional dissonances could presumably take
care of themselves. At least I don't know of any source, except the
Marpurg that I mentioned, that even once considers the question of
what the 'correct' tuning of functional dissonances should be.
~~~T~~~

Despite the endless looking into dyads, it is amazing there has not emerged any study that gives us a definitive answer to the question at hand. More in the lines of Bach three voice counterpuntal pieces, one notices when compared to Helmholtz's analysis of the inversions of triads, i noticed he favored the more dissonant ones. I think probably because it increases the independence of the the lines whereas as simple spacing might obscure it.

Tom Dent wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, > "Daniel Wolf" <djwolf@...> wrote:
> >
> > It's critical to distinguish between tonal dissonance and sensory
> > dissonance. Tonal dissonances are dissonant in terms of the musical
> > practice -- within the context of counterpoint and harmony -- while
> > sensory dissonance is a perceptual category. While the one may belong
> > more to a musical domain and the other to a physical domain, there
> is a
> > remarkable and useful overlap between these two descriptions
> (...)
>
> > in "well-composed" music of the style in question, augmented seconds,
> > augmented fifths, and diminished fourths will always be tonal
> dissonances,
> > even when they are acoustically identical -- of equal sensory
> consonance
> > -- to the minor thirds, minor sixths, and major thirds of another
> tonal
> > context.
> >
> > Daniel Wolf
> > Frankfurt
>
> That much is clear, but I still don't understand the question. Brad
> seems to want to take some advantage of the acoustical identity on the
> keyboard (at least the 12-note keyboard, which was by no means
> universal!) while on the other hand asking us to consider two-voiced
> compositions independently of any keyboard tuning.
>
> Two-voiced compositions, by definition, do not include a full harmony
> - until we have some sort of filling-in parts. If there is no
> complete harmony, the musical function of a tempered interval can
> often be ambiguous, relying on a wider harmonic context for a tonal
> 'explanation'. The A minor Prelude of Bach's 'Book 2' (24 Preludes and
> Fugues) is a possible example.
>
> Incidentally there are some 'classical-era' compositions for
> fixed-pitch instruments which are *not* "well-composed" in Daniel's
> sense: notably lute pieces by Silvius Leopold Weiss with enharmonic
> modulation via the augmented triad.
>
> The celebrated example of an acoustically consonant functional
> dissonance is the perfect fourth above the bass. (Other voicings of
> the fourth may be consonant...) No matter how perfectly in tune it is,
> it always required a resolution. So of course there is a distinction
> between functional dissonance in tonal style and acoustic dissonance.
> However, what this should have to do with tuning is not clear.
>
> Historically, in every source I know of in the 'classical' period, the
> aim of tuning is to make functional consonances sufficiently
> acoustically consonant. Functional dissonances could presumably take
> care of themselves. At least I don't know of any source, except the
> Marpurg that I mentioned, that even once considers the question of
> what the 'correct' tuning of functional dissonances should be.
> ~~~T~~~
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

> Posted by: "Tom Dent" stringph@gmail.com
> > I don't understand this supposed 'context'. Are there any pieces at
> all that satisfy the criteria Brad has demanded : exactly two voices
> and no influence from keyboard tuning?

Those weren't the criteria that I "demanded" or even "said", as to "no influence from keyboard tuning". I did say: "Disregard any and all issues of keyboard temperament." That is, consider for the moment a composition apart from any temperament(s) it might be played upon. How dissonant are augmented 5ths, functionally, as compared with minor 6ths playing EXACTLY the same pitch, however your keyboard may be tuned?

Want a good example of a composition that plays around with distinctions between augmented 5ths and minor 6ths, at close range? In exactly two-voiced texture? Bach's first Duetto, E minor, BWV 802.

Look at bars 3, 6, and 35. In each case we have D in the bass, and either Bb or A# in the treble being struck immediately after the D. From the context around these moments, bar 36 has (arguably) no dissonance at all in the D-Bb, but bar 3 which also has D-Bb seems at least mildly dissonant...probably under some influence from the enharmonic A# earlier in the same bar. And bar 6 is (again arguably) the most dissonant of all these three spots, as its A# over the D is within C-A#-B motion. Bar 23 is similar to 3, but in different octaves; less dissonant? More?

There's also apparently a game here where Bach is not only making enharmonic swaps as closely as possible (e.g. bars 3 RH, 9 LH), but also disguising parallel 5ths between the two voices as closely as possible (bars 2-4, et al).

Not to say that this Bach example is typical tonal music, though. I was wondering more about things that are less chromatic overall, yet occasionally have augmented 5ths or augmented 2nds or diminished 4ths showing up between melody and bass. The more diatonic a piece generally is, the bigger the dissonant effect when one of these augmented or diminished intervals comes up and shatters context...yes?

Try a simpler one. Goldberg Variations, variation 1, bar 18 (soon after the middle of the variation). We have a C in the bass several times, and a G# and later a D# above it: augmented 5th and augmented 2nd/9th. Now flip over to a minor variation, 15, and find a C-Eb such as in bar 26's downbeat, along with a C-Ab near the beginning of bar 25. Since this C-Ab and C-Eb are each within the scale of the moment, but the C-G# and C-D# are outside the scale of their moment, don't these augmented intervals seem a LOT more dissonant than their counterpart minor intervals?

How about "But who may abide" from Handel's "Messiah", bar 108ff, considering only the melody and bass? We're in D minor. The bass goes F-Bb, Bb-A. The violin has an A over the F, a C# and E over the first Bb, a D over the second Bb, and E-A over the A.

A, C# E D, E A.
F, Bb..Bb, A...

So, we're leaping in contrary motion to land on an excruciating Bb-C# augmented 2nd and then resolving it (properly) outward as the top voice moves up, leaping first to another dissonance (E over the Bb) and finally finding its D, a major 3rd above the bass. Pick some minor 3rds from anywhere else within this piece, and none of them will seem as dissonant as that Bb-C# augmented 2nd moment...will they?

Play through the melody and bass of Dido's Lament ("When I am laid in earth") from Purcell's "Dido and Aeneas". Doesn't it make its killer effect by having first an accented diminished 4th on the second bar, and then an augmented 2nd approached by parallel motion (a minor 3rd, and then both notes move down to make an augmented 2nd!) in the third bar?

G A Bb, Bb A Bn, C Bb A G F# G, F#.
G....G, F#...Fn, E........Eb.., D.

Brad Lehman