Tonal Affinities - Nature or Nurture?

At Ernst Terhardt's webpage on "Affinity of Tones":

http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/affinity.html
[http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/affinity.html]

Terhardt states:

<affinity. Affinity means that tones may be perceived as similar in certain
aspects; that in some respect a tone may be replaced by another one; and that
one tone may even be confused with another one. These criteria apply practically
only to two musical intervals, namely, theoctave, and, to a lesser degree, the
fifth. At least within the framework of tonal music, the affinity of tones being
in anoctave interval is so pronounced that this relationship is termedoctave
equivalence. For successive tones, affinity becomes apparent in the fact that
repetition of a musical phrase in a "transposition" by an octave, or, to a
lesser extent, by a fifth, is hardly perceived as a transposition at all - the
transposition is often hardly noticed. So, indeed, affinity is a kind of
similarity.>>

J Gill: Does it "ring true" (for you) that such "affinities" do "apply
practically only to two musical intervals, namely, the octave, and, to a lesser
degree, thefifth. "? Would you add other pairs of pitch intervals to (your)
list?

J Gill: Terhardt, on "octave and fifth matching of tones", states:

<template for the pitch interval that corresponds to an octave and fifth,
respectively.>>

J Gill: Expanding his discussion of possible other ratios for which certain
listeners may possess "auditory templates", Terhardt states:

<per se.) Rather, it appears likely that the pitch-interval templates have been
acquired, i.e., by passive and active "musical communication".>>

It appears (to me) that Terhardt (when the entire webpage is considered, in
addition to the brief "excerpts" from that webpage shown above), on balance,
leans toward a "nurture" (developed) as opposed to "nature" (born with)
viewpoint regarding the acquisition by persons of such "auditory templates".

What about people like Mozart? Particularly, what about "childhood progenies"
(where a minimum time over which to so aquire "auditory templates" exists). Do
such people simply inter-connect their neurons at rates which vastly exceed that
of an average child? Are they more likely to explore the tonal "possibilities"
at a higher rate, thus "learning" faster?

Would you agree that it is true that "musical abilities" (in general) are not
(at all) related to "nature" ("in-born")? Is it all "perspiration", without (any
"in- born) "inspiration"? Therefore, does musical "inspiration" arise only out
of "perspiration". How does this explain the (often heard) sentiments of
musicians relating to the musical "improvisation" of musicians, to the effect
that - rather than their concious minds speaking *through the music*, the
"music" seems, instead to "speak" *through the musician* (without a focal
concious awareness at play on the part of the practicing/performing musician
during the process of the action of creating sounds).

Note that Terhardt (in the full text of his webpage) refers to his concepts of
"affinity of tones" as relating to succesive tones ("melody") as well as
simultaneous tones ("harmony").

Curiously, J Gill

In-Reply-To: <9vnnqj+tdls@eGroups.com>
jacky wrote:

> With all due respect to dear Mr. Terhardt, I would have to say that
> even though he does not seem to mention timbre here *at all*, he must
> have some special timbres in mind.

Nope. If you read down the page, it says "Octave- and fifth-matching can
be done with practically any type of tone, in particular, both with
harmonic complex tones and with sine tones." This is from:

<http://www.mmk.ei.tum.de/persons/ter/top/affinity.html>

Graham

In-Reply-To: <5.1.0.14.2.20011218063616.00a1c7f0@mail.imajis.com>
J Gill wrote:

> J Gill: Does it "ring true" (for you) that such
> "affinities" do "apply practically only to two musical
> intervals, namely, the /octave/, and, to a lesser degree, the
> /fifth/. " Would you add other pairs of pitch intervals to
> (your) list?

Yes, it rings true. I wouldn't add or subtract anything without
experimental data to back it up. But octaves and fifths do have a special
place in music, so perhaps this is why.

> It appears (to me) that Terhardt (when the entire webpage is considered,
> in addition to the brief "excerpts" from that webpage shown
> above), on balance, leans toward a "nurture" (developed) as
> opposed to "nature" (born with) viewpoint regarding the
> acquisition by persons of such "auditory templates".

It depends on what you consider the "normal" position. I think he's
leaning towards "nature" in even allowing octaves and fifths to be
"natural" for melody.

> What about people like Mozart? Particularly, what about "childhood
> progenies" (where a minimum time over which to so aquire
> "auditory templates" exists). Do such people simply
> inter-connect their neurons at rates which vastly exceed that of an
> average child? Are they more likely to explore the tonal
> "possibilities" at a higher rate, thus "learning"
> faster?

I think that's a complete red herring. Mozart was the son of a
professional composer, of course he would have been exposed to more
diatonic music and so required less time to acquire the templates for
diatonic intervals. Do you mean prodigies? There must be more to musical
talent than simply recognising melodic intervals. I don't even see
anything in <http://www.mmk.ei.tum.de/persons/ter/top/affinity.html> about
what age these templates are normally acquired. Perhaps all children with
exposure to diatonic music get them by age 5 anyway.

> Would you agree that it is true that "musical abilities" (in
> general) are not (at all) related to "nature"
> ("in-born")?

Well, that's a can of hornets' nests I'd prefer not to open.

Graham

>At Ernst Terhardt's webpage on "Affinity of Tones":

>http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/affinity.html

>Terhardt states:

><<In tonal music, a number of fundamental auditory characteristics
>of tones play an important role, and one of these characteristics is
>affinity. Affinity means that tones may be perceived as similar in
>certain aspects; that in some respect a tone may be replaced by
>another one; and that one tone may even be confused with another
>one. These criteria apply practically only to two musical intervals,
>namely, the octave, and, to a lesser degree, the fifth. At least
>within the framework of tonal music, the affinity of tones being in
>an octave interval is so pronounced that this relationship is termed
>octave equivalence. For successive tones, affinity becomes apparent
>in the fact that repetition of a musical phrase in a "transposition"
>by an octave, or, to a lesser extent, by a fifth, is hardly
>perceived as a transposition at all - the transposition is often
>hardly noticed. So, indeed, affinity is a kind of
>similarity.>>

>J Gill: Does it "ring true" (for you) that such "affinities"
>do "apply practically only to two musical intervals, namely, the
>octave, and, to a lesser degree, the fifth. "? Would you add other
>pairs of pitch intervals to (your) list?

Well, I'd add the fourth, but probably Terhardt already had this
included, since it's merely an octave-affinity combined with a fifth-
affinity.

This rings VERY TRUE to me. Note that Terhardt is talking about
_successive tones_ -- not _simultaneous tones_. In a statistical
sampling of musical scales in cultures where simultaneous harmony is
not used, one sees only very few intervals show up with more
frequency than one could attribute to chance. These intervals are the
octave, fifth, and fourth.

>What about people like Mozart? Particularly, what about "childhood
>progenies" (where a minimum time over which to so aquire "auditory
>templates" exists). Do such people simply inter-connect their
>neurons at rates which vastly exceed that of an average child? Are
>they more likely to explore the tonal "possibilities" at a higher
>rate, thus "learning" faster?

Sounds reasonable -- but note that Terhardt does not take such a
strong "nurture over nature" view when it comes to _simultaneous
harmony_.

>Would you agree that it is true that "musical abilities" (in
>general) are not (at all) related to "nature" ("in-born")?

Not at all, and that's not at all what Terhardt is arguing here or in
most of the other webpages on his site.

>How does this explain the (often heard) sentiments of musicians
>relating to the musical "improvisation" of musicians, to the effect
>that - rather than their concious minds speaking *through the
>music*, the "music" seems, instead to "speak" *through the musician*
>(without a focal concious awareness at play on the part of the
>practicing/performing musician during the process of the action of
>creating sounds).

I don't think that those who feel this way (myself included) would
tend to or feel logically compelled to fall into either the "nature"
or "nurture" camps when it comes to the origin of musical tone-
systems. On second thought, it seems quite evident to me that even
the deepest and most profound improvisational expressions of the
musicians of a given culture are close to _totally unintelligible_ to
listeners from a greatly removed culture, almost as profoundly as a
great work of literature in the Anaphorian language (my apologies,
Kraig) would be totally unintelligible to you, an English speaker.

>Note that Terhardt (in the full text of his webpage) refers to his
>concepts of "affinity of tones" as relating to succesive tones >
("melody") as well as simultaneous tones ("harmony").

True, but he is not nearly as restrictive when it comes to the set of
intervals that have "special characteristics" such as "consonance" in
the use of simultaneous tones. "Affinity" is only one possible
characteristic that a pair of tones may have, but its the only one
that seems to occur "in nature" for _successive tones_. Other "non-
affine" pairs of tones may nevertheless possess a _special
relationship_, independent of _nurture_, when it comes to
_simultaneous tones_. (Note that when I say "in nature", I'm
considering intimate, unconscious familiarity with the spectrum of
the human voice "natural", since hardly any musician in the world can
be said to operate without such familiarity, whether it's inborn or
not.)

--- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:

> With all due respect to dear Mr. Terhardt, I would have to say that
> even though he does not seem to mention timbre here *at all*, he
must
> have some special timbres in mind.

Actually, he states that the effect hold good even with sine waves.

> To speak about this kind of thing
> outside of a musical context, where timbre, rhythm, and spacial
> considerations are removed, is only looking at the *surface* of
> things. When one considers the range of possible fifths and near
> octaves, then put that into a real world musical situation - the
> context will reveal if this is so or not. It's a little hard to
> imagine that this has relevance to every musical situation possible
> on the globe.

Jacky -- I think the power of Terhardt's observation here is in
explaining the fact that virtually all musical cultures utilize
intervals near to octaves, fifths, or fourths, while all other
intervals occur with no such universality. One finds second, thirds,
and unclassifiable intervals of all sorts in various musical
cultures, and there doesn't seem to be any tendency to share those
same interval choices, or avoid the same non-choices, from one
culture to another. Only within a culture do a given fixed set of
second, thirds, and/or other intervals categories seem to acquire the
status of "correct" and "incorrect" -- except the octave, fifth, and
fourth which, while subject to "flavoring", do appear to occur
universally. Wouldn't you agree?

--- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> --- In tuning@y..., graham@m... wrote:
> and with sine tones."
>
> And pray tell, what do sine tones have to do with musical timbres?
>
> Point me to a composer who's composing music with sines. I'd enjoy
> hearing their work.
>
> JL

David Beardsley, La Monte Young, . . .

jacky_ligon@yahoo.com (jacky_ligon) wrote:

> --- In tuning@y..., graham@m... wrote:
> and with sine tones."

Jacky, the four words you quoted there aren't from me at all, but Ernst
Terhardt. If you have a problem, take it up with him. The website gives
his e-mail address as terhardt@ei.tum.de

> Point me to a composer who's composing music with sines. I'd enjoy
> hearing their work.

If you mean Pan[a]sonic, they're a duo. Certainly well worth listening
to, but make sure you get the right CD as they didn't keep up with the
pure sine approach. I could look up the title if you like, but not at
this time of night.

Graham

Paul Erlich wrote:

> Well, I'd add the fourth, but probably Terhardt already had this
> included, since it's merely an octave-affinity combined with a fifth-
> affinity.

No, he says "I believe one can be sure that any person with some musical
talent and training is able to match a few more intervals as well, in
particular, the major and minor seconds, the major and minor thirds, and
the fourth."

Graham

--- In tuning@y..., graham@m... wrote:
> jacky_ligon@y... (jacky_ligon) wrote:
>
> > --- In tuning@y..., graham@m... wrote:
> > and with sine tones."
>
> Jacky, the four words you quoted there aren't from me at all, but
Ernst
> Terhardt. If you have a problem, take it up with him. The website
gives
> his e-mail address as terhardt@e...
>
> > Point me to a composer who's composing music with sines. I'd
enjoy
> > hearing their work.
>
> If you mean Pan[a]sonic, they're a duo. Certainly well worth
listening
> to, but make sure you get the right CD as they didn't keep up with
the
> pure sine approach. I could look up the title if you like, but not
at
> this time of night.
>
>
> Graham

Of course any composer who uses additive synthesis is, in
reality, "composing music with sines".

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, December 18, 2001 12:16 PM
> Subject: [tuning] Re: Prime Rarity?
>
>
> When it comes to _intervals_, primeness or non-primeness is
> _unimportant_ in describing the effect of the interval.
> I agree with Dave Keenan that the three factors of importance are
>
> (1) COMPLEXITY -- the size of the numbers in the ratio
> (2) TOLERANCE -- the inability of the ear to distinguish slight
> mistuning
>
> [(1) & (2) are combined in a more sophisticated way in harmonic
> entropy]
>
> (3) SPAN -- the actual *size* (as in cents) of the interval.

Just thought I'd put it on the record that I agree with this too.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, December 18, 2001 1:58 PM
> Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
>
>
> Jacky -- I think the power of Terhardt's observation here is in
> explaining the fact that virtually all musical cultures utilize
> intervals near to octaves, fifths, or fourths, while all other
> intervals occur with no such universality. One finds second, thirds,
> and unclassifiable intervals of all sorts in various musical
> cultures, and there doesn't seem to be any tendency to share those
> same interval choices, or avoid the same non-choices, from one
> culture to another. Only within a culture do a given fixed set of
> second, thirds, and/or other intervals categories seem to acquire the
> status of "correct" and "incorrect" -- except the octave, fifth, and
> fourth which, while subject to "flavoring", do appear to occur
> universally. Wouldn't you agree?

*This*, however, is a perfect example of the kind of situation
where I think primeness plays a role.

AFAIK, and alongside the points Paul makes here, the only
"intervals of equivalence" that have ever been observed in the
kind of musical practices which develop outside of written
theory, are the 8ve, 5th, and 4th.

(Examples of many other types of intervals of equivalence can be
found in the scales being generated over on the Tuning-Math list.)

I would argue that the reasons *why* those particular intervals
have been chosen, all around the earth, time and time again, are
because of properties of affinity which result from the low
primeness of these intervals:

where r = 2^x * 3^y,

r x y

8ve ( 1 0 )
5th (-1 1 )
4th ( 2 -1 )

In terms of 3-limit prime-factorization, these are the three
simplest intervals -- after ( 0 0 ), which is the unison.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:

/tuning/topicId_unknown.html#31645
>
> With all due respect to dear Mr. Terhardt, I would have to say that
> even though he does not seem to mention timbre here *at all*, he
must
> have some special timbres in mind. To speak about this kind of
thing
> outside of a musical context, where timbre, rhythm, and spacial
> considerations are removed, is only looking at the *surface* of
> things. When one considers the range of possible fifths and near
> octaves, then put that into a real world musical situation - the
> context will reveal if this is so or not. It's a little hard to
> imagine that this has relevance to every musical situation possible
> on the globe.
>
> JL

Hi Jacky!

This reminds me a little of the experiment that Bill Sethares does in
his _Tuning, Timbre, Spectrum, Scale_ where he adjusts the timbre and
makes a min. 9th (is it a 9th, I don't have the book right here)
sound like an octave, and the *true* octave sounds like a
dissonance. It's a great experiment and one of the first things in
the book... and one of the most memorable...

Joseph

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., graham@m... wrote:
> > jacky_ligon@y... (jacky_ligon) wrote:
> >
> > > --- In tuning@y..., graham@m... wrote:
> > > and with sine tones."
> >
> > Jacky, the four words you quoted there aren't from me at all, but
> Ernst
> > Terhardt. If you have a problem, take it up with him. The
website
> gives
> > his e-mail address as terhardt@e...
> >
> > > Point me to a composer who's composing music with sines. I'd
> enjoy
> > > hearing their work.
> >
> > If you mean Pan[a]sonic, they're a duo. Certainly well worth
> listening
> > to, but make sure you get the right CD as they didn't keep up
with
> the
> > pure sine approach. I could look up the title if you like, but
not
> at
> > this time of night.
> >
> >
> > Graham
>
> Of course any composer who uses additive synthesis is, in
> reality, "composing music with sines".

Bob Wendell answers:

Then by this implicit, operative definition everybody who uses any
kind of periodic waveform is "composing with sines" and the
terminology becomes totally meaningless. All periodic waveforms are
composite sine-wave structures per Fourier.

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> I would argue that the reasons *why* those particular intervals
> have been chosen, all around the earth, time and time again, are
> because of properties of affinity which result from the low
> primeness of these intervals:

I would argue that it has nothing to do with the "low primeness", but
rather with the fact that these are the simplest possible ratios, the
ratios that involve the smallest numbers.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, December 19, 2001 11:18 AM
> Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > I would argue that the reasons *why* those particular intervals
> > have been chosen, all around the earth, time and time again, are
> > because of properties of affinity which result from the low
> > primeness of these intervals:
>
> I would argue that it has nothing to do with the "low primeness", but
> rather with the fact that these are the simplest possible ratios, the
> ratios that involve the smallest numbers.

Hmmm... well, I can go along with you to some extent.

But 5 is only one more than 4, so why is there so much difference
and inconsistency in the way the world's musical cultures have
treated 5-limit intervals, when there's so much similarity and
consistency with the way those with factors of 2 and 3 have been
treated? Clearly, there is some very special difference between
3-limit and 5-limit ratios, more than simply the size of the numbers.

-monz

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--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

> > Of course any composer who uses additive synthesis is, in
> > reality, "composing music with sines".
>
> Bob Wendell answers:
>
> Then by this implicit, operative definition everybody who uses any
> kind of periodic waveform is "composing with sines" and the
> terminology becomes totally meaningless. All periodic waveforms are
> composite sine-wave structures per Fourier.

Granted, but I would not consider it "composing with sines" unless
the composer had specified exactly which Fourier components would
appear with exactly what relative amplitudes, as is the case in
additive synthesis.

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Wednesday, December 19, 2001 11:18 AM
> > Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > I would argue that the reasons *why* those particular intervals
> > > have been chosen, all around the earth, time and time again, are
> > > because of properties of affinity which result from the low
> > > primeness of these intervals:
> >
> > I would argue that it has nothing to do with the "low primeness",
but
> > rather with the fact that these are the simplest possible ratios,
the
> > ratios that involve the smallest numbers.
>
>
> Hmmm... well, I can go along with you to some extent.
>
> But 5 is only one more than 4, so why is there so much difference
> and inconsistency in the way the world's musical cultures have
> treated 5-limit intervals, when there's so much similarity and
> consistency with the way those with factors of 2 and 3 have been
> treated? Clearly, there is some very special difference between
> 3-limit and 5-limit ratios, more than simply the size of the
numbers.
>
>
> -monz

Terhardt did _not_ include the fourth in his list of "affine"
intervals. I tend to include it, but only because I tend to assume
octave-equivalence, and in an octave-equivalent view of a musical
system, an affinity at a fifth translates into an affinity at a
fourth. Even if one does not explicitly assume octave-equivalence,
one can argue that, since virtually all cultures have experimented
with building both an octave and a fifth in the same direction from a
given starting pitch, the vast majority of those would have stumbled
upon the fourth early and often in their constructions, and hence it
would acheive a measure of ubiquity.

Similarly, a view which is focused on odd limit rather than prime
limit would address your objection very well, since an odd limit of 3
would include only the "affine" intervals, while a prime limit of 3
would also include intervals such as 9:8, 16:9, 27:16, 32:27, 81:64,
etc.

I guess these arguments are not as convincing as the simple statement
that while primeness is an important property of _pitch systems_, it
has absolutely nothing to say about the innate, non-cultural, sensual
quality of an interval, when generalized away from any musical
context. This may seem overly abstract to you, but it seems essential
to me that one eliminates unnecessary hypotheses from a theory, and
ascribing importance to "primeness" in isolated intervals seems just
such a hypothesis, especially if you're going to build up from those
isolated intervals and continue to have a clear picture of what's
going on. Primeness comes into the picture when you look at JI tuning
systems precisely inasmuch as the mathematics of primes (I'm thinking
the prime factorization theorem) forces it to do so, and no more and
no less, I would argue. I see no evidence to the contrary, and
ascribing a "prime sense" to our ears seems like a clear violation of
Occam's razor to me.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> Terhardt did _not_ include the fourth in his list of "affine"
> intervals.

Affine? What could make an interval affine?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Terhardt did _not_ include the fourth in his list of "affine"
> > intervals.
>
> Affine? What could make an interval affine?

J Gill: As Stan Laurel used to say to Oliver Hardy,

"Affine mess you've gotten us into *this* time, Ollie!".

JG :)

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Terhardt did _not_ include the fourth in his list of "affine"
> > intervals.
>
> Affine? What could make an interval affine?

Did you read the article we're talking about?

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> >
> > > Terhardt did _not_ include the fourth in his list of "affine"
> > > intervals.
> >
> > Affine? What could make an interval affine?
>
>
> J Gill: As Stan Laurel used to say to Oliver Hardy,
>
> "Affine mess you've gotten us into *this* time, Ollie!".
>
>
> JG :)

Bob Wendell:
Ha-ha! Gotta love it!!!

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> > > Of course any composer who uses additive synthesis is, in
> > > reality, "composing music with sines".
> >
> > Bob Wendell answers:
> >
> > Then by this implicit, operative definition everybody who uses
any
> > kind of periodic waveform is "composing with sines" and the
> > terminology becomes totally meaningless. All periodic waveforms
are
> > composite sine-wave structures per Fourier.
>
> Granted, but I would not consider it "composing with sines" unless
> the composer had specified exactly which Fourier components would
> appear with exactly what relative amplitudes, as is the case in
> additive synthesis.

Bob answers:
Well, I see the point technically, but it seems a bit semantically
specious, since whether by intentional composition of tones or
compositional choice of tones, it is composition, and all periodic
waveforms are built of sine-wave components. I do not infer from my
reading of the post that spawned this issue that "composing with
sines" was intended this way.

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

> Bob answers:
> Well, I see the point technically, but it seems a bit semantically
> specious, since whether by intentional composition of tones or
> compositional choice of tones, it is composition, and all periodic
> waveforms are built of sine-wave components. I do not infer from my
> reading of the post that spawned this issue that "composing with
> sines" was intended this way.

I would argue that it may be sematically specious to attempt to draw
a line here, since those who compose with sine waves may in fact,
intentionally or otherwise, blur the boundaries between using sine
waves as timbres and using sine waves as the constituents of timbres.
I would blur those boundaries, anyway, if I were to attempt
to "compose with sine waves".

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_31643.html#31668

>
> Jacky -- I think the power of Terhardt's observation here is in
> explaining the fact that virtually all musical cultures utilize
> intervals near to octaves, fifths, or fourths, while all other
> intervals occur with no such universality. One finds second,
thirds,
> and unclassifiable intervals of all sorts in various musical
> cultures, and there doesn't seem to be any tendency to share those
> same interval choices, or avoid the same non-choices, from one
> culture to another. Only within a culture do a given fixed set of
> second, thirds, and/or other intervals categories seem to acquire
the
> status of "correct" and "incorrect" -- except the octave, fifth,
and
> fourth which, while subject to "flavoring", do appear to occur
> universally. Wouldn't you agree?

Let me make sure I understand this interesting discussion... When you
speak of the "affinity," Paul... in Terhardt's world is this
universality of the octave, fifth and fourth strictly a *melodic*
consideration in this context??

P.S. I should read the Terhardt again, I admit...

Joseph

> Let me make sure I understand this interesting discussion... When
you
> speak of the "affinity," Paul... in Terhardt's world is this
> universality of the octave, fifth and fourth strictly a *melodic*
> consideration in this context??

It's octave and fifth, and it's both melodic and harmonic. Other
intervals may be harmonically consonant, but don't give you the sense
that "the two pitches are really similar".

> > Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > I would argue that the reasons *why* those particular intervals
> > > have been chosen, all around the earth, time and time again, are
> > > because of properties of affinity which result from the low
> > > primeness of these intervals:
> >
> > I would argue that it has nothing to do with the "low primeness", but
> > rather with the fact that these are the simplest possible ratios, the
> > ratios that involve the smallest numbers.
>
>
> Hmmm... well, I can go along with you to some extent.
>
> But 5 is only one more than 4, so why is there so much difference
> and inconsistency in the way the world's musical cultures have
> treated 5-limit intervals, when there's so much similarity and
> consistency with the way those with factors of 2 and 3 have been
> treated? Clearly, there is some very special difference between
> 3-limit and 5-limit ratios, more than simply the size of the numbers.

Or look at meter. 3 and 2, and their "octave equivalents, 4, 6, 8 (and
to a lesser extent 9 and 12) dominate musics which have a metric
orientation. The more comlicated prime rhythm musics (Balkan and Indian
musics) tend to heirarchically arrive at those primes by summing 2's, 3's
and 4's.

Its not just that small numbers are "good", but very small numbers are
"good". As has been pointed out, due to the definition of prime numbers,
the population of primes is more dense in the population of small
numbers.

Bob Valentine

> From: Robert C Valentine <BVAL@IIL.INTEL.COM>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, December 19, 2001 11:43 PM
> Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
>
>
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > But 5 is only one more than 4, so why is there so much difference
> > and inconsistency in the way the world's musical cultures have
> > treated 5-limit intervals, when there's so much similarity and
> > consistency with the way those with factors of 2 and 3 have been
> > treated? Clearly, there is some very special difference between
> > 3-limit and 5-limit ratios, more than simply the size of the numbers.
>
> Or look at meter. 3 and 2, and their "octave equivalents, 4, 6, 8 (and
> to a lesser extent 9 and 12) dominate musics which have a metric
> orientation. The more comlicated prime rhythm musics (Balkan and Indian
> musics) tend to heirarchically arrive at those primes by summing 2's, 3's
> and 4's.
>
> Its not just that small numbers are "good", but very small numbers are
> "good". As has been pointed out, due to the definition of prime numbers,
> the population of primes is more dense in the population of small
> numbers.

In fact, Bob, I got many of my early ideas about primeness from
exactly this! I've been a huge fan of Bulgarian folk music for about
20 years now, and my own explorations into the asymmetrical meters
which are common in that repertoire led me to the same realization.

All complex meters are most easily comprehended by breaking them
up into smaller units until everything is compounded of basic units
of 2 or 3 beats.

I began to understand rational harmony in the same way, and it
seemed to me that in a very general sense, Partch was on to something
with his idea that there can be a chronological model of the
acceptance by various world cultures of successively higher primes
as 1) harmonic identities, and 2) consonant intervals.
I made that idea the core of my own book.

In any case, it seems clear to me that humans "understand" many
different aspects of numerical grouping in music (i.e., both
harmonic and metric) by means of the relationships of 3-limit ratios.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> I began to understand rational harmony in the same way, and it
> seemed to me that in a very general sense, Partch was on to
something
> with his idea that there can be a chronological model of the
> acceptance by various world cultures of successively higher primes
> as 1) harmonic identities, and 2) consonant intervals.
> I made that idea the core of my own book.

Check Partch again. He clearly sees this model as applying to *odd*s,
not to *primes*. Every single discussion of harmonic identities and
of consonance, except for the single half-page entitles "The Enigma
of the Multiple-Number Ratio", backs me up on this.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, December 20, 2001 12:02 PM
> Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > I began to understand rational harmony in the same way, and it
> > seemed to me that in a very general sense, Partch was on to
> > something with his idea that there can be a chronological model
> > of the acceptance by various world cultures of successively
> > higher primes as 1) harmonic identities, and 2) consonant
> > intervals. I made that idea the core of my own book.
>
> Check Partch again. He clearly sees this model as applying to *odd*s,
> not to *primes*. Every single discussion of harmonic identities and
> of consonance, except for the single half-page entitles "The Enigma
> of the Multiple-Number Ratio", backs me up on this.

Oops... my bad. Of course you're right about this, Paul. *I*
transferred Partch's idea to apply to primes! ... and others
before me did too, yes? The first thing I ever read about
prime-affect was a paper by Scott Makeig, who (I later found out)
got the idea from Danielou.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com