Re-opening the theories of consonance and dissonance.

> The conclusion was that ratios of small whole numbers are
> more consonant
> Largely dismissed were metrics based on prime factorization.

If prime factorization was discounted, how do you account for inversions?
4/3 is every bit as consonant as 3/2, but the numbers involved are almost
50% larger. Isn't that where the prime number theory came in?

> adapted to explain this by including a TOLERANCE function. Everybody's
> favorite is Paul Erlich's Harmonic Entropy, which, along the
> way, explains
> one of the reasons why small-numbered ratios are consonant in
> the first place.

I'm really sorry I missed this one. Could someone repost Harmonic Entropy?

> 2. SPAN- When intervals are very small, they can be highly dissonant
> example, one can play almost any notes together 6 octaves
> apart on a piano
> and they won't sound particularly consonant, but they won't
> clash either.

William A. Sethares suggests in his book, Tuning Timbre, Spectrum, Scale
that the structure of partials of a tone determines what other notes it will
be consonant with. The spectra of sound samples he used was too much like
multiple notes played at once and failed to convince me of his point.

However, your example of the notes far apart on the piano is much more
suggestive that the harmonic spectra spectra of the two notes effect our
perception of their consonance. Maybe the notes being so far apart makes
the spectra so unrelated, that we can no longer make sense of what we are
hearing. How else would you explain this phenomenon?

On a related note, Is it possible to produce a pure sound of a sine wave
with no additional partials whatsoever? Even the sound resonating in our
hearing mechanisms would produce some partials. Wouldn't it?

---
Glen Peterson
30 Elm Street North Andover, MA 01845
(978) 975-1527
http://www.OrganicDesign.org/Glen/Instruments

about 4/3 being just as consonant as 3/2... some scholars had a different
opinion. Some scholars of counterpoint consider the 4th dissonant. And
some consider is consonant, but not as consonant as the Perfect 5th. In
voice leading, you may have parallel 4ths but not parallel 5ths or
octaves....

My 2^(1/600)
Alex

No Jive!

That's why the whole low-ratios-are-consonant argument
never has held water. It totally ignores the fourth.
(paraphrase of Charles)

Rick S.
Manhattan

> From: "Azi of Vajravai Mo'f'ck Mage" <vajravai@hotmail.com>
>
> about 4/3 being just as consonant as 3/2... some scholars had a different
> opinion. Some scholars of counterpoint consider the 4th dissonant. And
> some consider is consonant, but not as consonant as the Perfect 5th. In
> voice leading, you may have parallel 4ths but not parallel 5ths or
> octaves....

>If prime factorization was discounted, how do you account for inversions?

Eh?

>4/3 is every bit as consonant as 3/2, but the numbers involved are almost
>50% larger. Isn't that where the prime number theory came in?

I don't think 4/3 is as consonant as 3/2. The geometric mean metric gives
the former a dissonance of 3.46 and the latter a dissonance of 2.45, which
jives real good with my experience.

>I'm really sorry I missed this one. Could someone repost Harmonic Entropy?

Check http://www.uq.net.au/~zzdkeena/Erlich/index.htm or search the list
archives on the onelist site.

>http://www.OrganicDesign.org/Glen/Instruments

Nice!

>That's why the whole low-ratios-are-consonant argument
>never has held water. It totally ignores the fourth.

I think the sound of the 4/3 is measured quite well by the metrics
mentioned. However, your language reveals something else may be at work-
the 4/3 is the fourth degree of a very popular scale, and its function
there may contribute to your perception of it.

-C.

>From: "Azi of Vajravai Mo'f'ck Mage" <vajravai@hotmail.com>
>Reply-To: tuning@onelist.com
>To: tuning@onelist.com
>Subject: Re: [tuning] Re-opening the theories of consonance and dissonance.
>Date: Tue, 24 Aug 1999 15:55:45 GMT
>
>From: "Azi of Vajravai Mo'f'ck Mage" <vajravai@hotmail.com>
>
>about 4/3 being just as consonant as 3/2... some scholars had a different
>opinion. Some scholars of counterpoint consider the 4th dissonant. And
>some consider is consonant, but not as consonant as the Perfect 5th. In
>voice leading, you may have parallel 4ths but not parallel 5ths or
>octaves....
AS A COMPOSER I CAN ASSURE YOU THAT PARALLEL 4THS, 5THS, SECONDS, THIRDS,
TRITONES, MINOR SECONDS, AD INFINITUM ARE TOTALLY PERMISSABLE. IT IS ONLY
THE BEGINNING STUDENT THAT NEEDS TO ADHERE TO THE "BOOK".
>My 2^(1/600)
>Alex
>
>
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Glen Peterson wrote,

>If prime factorization was discounted, how do you account for inversions?
>4/3 is every bit as consonant as 3/2, but the numbers involved are almost
>50% larger. Isn't that where the prime number theory came in?

No sir. Although I would say that 3/2 is more consonant than 4/3, in my
usual work in music theory I treat them as equally consonant, since they are
octave-equivalent. That is because they both have an ODD limit, or greatest
odd factor, of 3. In the context of octave-equivalence, odd limit (up to
about 11 or 13) is a fine measure of consonance, better than anything that
incorporates prime factorization. An examination of Harry Partch's
"One-Footed Bride" reveals that he used odd limit as a measure of
consonance.

>I'm really sorry I missed this one. Could someone repost Harmonic Entropy?

Some of my thoughts are collected at
http://www.ixpres.com/interval/td/entropy.htm; the same thing plus
commentary by Joe Monzo and six graphs by me is at
http://www.uq.net.au/~zzdkeena/Erlich/index.htm.

>William A. Sethares suggests in his book, Tuning Timbre, Spectrum, Scale
>that the structure of partials of a tone determines what other notes it
will
>be consonant with. The spectra of sound samples he used was too much like
>multiple notes played at once and failed to convince me of his point.

Agreed. We seem to have a built-in ability to recognize harmonic series, but
not other partial-structures, as a single note. That is the basis of
harmonic entropy.

>On a related note, Is it possible to produce a pure sound of a sine wave
>with no additional partials whatsoever? Even the sound resonating in our
>hearing mechanisms would produce some partials. Wouldn't it?

Yes (though I wouldn't call it resonance). If our brains were somehow
stimulated to produce the impression of a pure sine wave, it would be
uinlike anything that we could actually hear through our ears.

-Paul

[Rick Sanford:]
> No Jive!
>
> That's why the whole low-ratios-are-consonant argument
> never has held water. It totally ignores the fourth.

I think it's safer to say that this offers a cautionary example of why
(to paraphrase Rick) the whole low-ratios-are-better (better for music
that is) argument is something a lot less than airtight... Of course
it may often be the case that "low-ratios" are "better" for music, but
it just as well (if perhaps less often) may not be so.

The obvious equilibrium tuning and music share seems to create an
extremely inveigling condition, where explaining (or understanding)
both under the umbrella of a (vis-a-vis) 'convertible philosophy'
overrides any evidential, or cautionary objection that seems to say:
"...Music and tuning are not the same..." which is only to say that
while tuning and music may very well (tranquilly and prudently) adhere
to the same persuasive conditions of right and wrong, one may also be
acutely conspicuous for its (stubborn and brutal) anachronistic stance
in lieu of the others 'fiats and literatim.'

Dan

I forgot to mention that Carl's summary inspired more thoughts than just
about any post I have read so far. Very clearly written too.

I am slowly making my way through "on Harmonic Entropy by Paul Erlich"

http://www.uq.net.au/~zzdkeena/Erlich/index.htm

First off, Great stuff! The graphs are wonderful. Has anyone made a
similar graph by playing intervals and asking people how consonant they
sound? Anyone interested in making an online test to do just such a thing?

Here are some quotes from the site with my comments:

"Recent speculation among tuning theorists has raised the idea that
consonance and dissonance may actually be two separate and not
mutually-exclusive dimensions of sonance. I have extrapolated this to the
idea that each prime may in fact be responsible for a separate dimension of
sonance that does not necessarily exclude any of the others."

What about saying each odd factor instead of each prime? I have found in my
experiments that n/3 will be consonant with n/6 and n/9, but n/6 and n/9 are
not so consonant with each other. All are related by a factor 3, but the
relationship doesn't carry over between 2*3 and 3*3.

"For triads or higher-ads - the roughness of Otonal chords is the same as
that of Utonal chords because they have the same intervals, but the
tonalness of Otonal chords is much greater, because they imply a much
simpler set of harmonics over a fundamental."

Also, difference tones from Otonal dyads are generally consonant, while
Utonal dyads often produce dissonant ones.

> From: Carl Lumma <clumma@nni.com>
> >If prime factorization was discounted, how do you account
> >for inversions?
> Eh?

I was thinking about a "fifth" up and a "fourth" down as being the same. It
makes sense that in actual music, the intervals are heard differently. I
have also been wondering if I even consider "fourths and fifths" to be
consonant or dissonant. They are almost neither.

---
Glen Peterson
30 Elm Street North Andover, MA 01845
(978) 975-1527
http://www.OrganicDesign.org/Glen/Instruments

Glen wrote,

>I am slowly making my way through "on Harmonic Entropy by Paul Erlich"

>http://www.uq.net.au/~zzdkeena/Erlich/index.htm

[...]

>Here are some quotes from the site with my comments:

>"Recent speculation among tuning theorists has raised the idea that
>consonance and dissonance may actually be two separate and not
>mutually-exclusive dimensions of sonance. I have extrapolated this to the
>idea that each prime may in fact be responsible for a separate dimension of
>sonance that does not necessarily exclude any of the others."

>What about saying each odd factor instead of each prime? I have found in my
>experiments that n/3 will be consonant with n/6 and n/9, but n/6 and n/9 are
>not so consonant with each other. All are related by a factor 3, but the
>relationship doesn't carry over between 2*3 and 3*3.

I totally agree and would say "odd" where Joe Monzo said "prime". Unfortunately
it is not so clear on the site which is my original text and which is Joe's
commentary. Joe, perhaps it would be helpful to put the initials of the author
in front of each paragraph?

Just a comment. Berio made it very clear that the P4 had to used carefully or
not at all. He mentioned to his composition students that orchestrating it led
to some disturbing and unexpected clashes in the harmonics of the two notes.
RT

Rick Sanford wrote:

> From: Rick Sanford <rsanf@pais.org>
>
> No Jive!
>
> That's why the whole low-ratios-are-consonant argument
> never has held water. It totally ignores the fourth.
> (paraphrase of Charles)
>
> Rick S.
> Manhattan
>
> > From: "Azi of Vajravai Mo'f'ck Mage" <vajravai@hotmail.com>
> >
> > about 4/3 being just as consonant as 3/2... some scholars had a different
> > opinion. Some scholars of counterpoint consider the 4th dissonant. And
> > some consider is consonant, but not as consonant as the Perfect 5th. In
> > voice leading, you may have parallel 4ths but not parallel 5ths or
> > octaves....
>
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Rick Tagawa wrote,

>Just a comment. Berio made it very clear that the P4 had to used carefully
or
>not at all. He mentioned to his composition students that orchestrating it
led
>to some disturbing and unexpected clashes in the harmonics of the two
notes.

It's more a stylistic constraint than anything else. In pop and rock the
perfect fourth is very common and is as smooth a consonance as any other.