Pitching in for "Pitch"

Hi Monz, Carl,

For definition of Pitch: "The subjective sensation of frequency".

I thought of adding 'subjective', because frequency and pitch may not be synonymous.

How does it matter if the definition is succinct? In fact, the fewer the terms used to define a term, the simpler it would be.

Further, the expanantory part of a definition should be separated [call it note, explanation, addendum, comment ......], keeping the definition uncluttered.

I am a little concerned about the term "Interval" -- I will refer to it later.

Regards,
Haresh.

>I am a little concerned about the term "Interval" -- I will refer to it
>later.

Monzos def: "The size between two different pitches, or the section of
the linearly-perceived pitch-continuum bounded by those two pitches."

I'd say distance, not size.

David Beardsley

--- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
............

>>>> Monzos def: "The size between two different pitches, or the section of the linearly-perceived pitch-continuum bounded by those two pitches."
> > I'd say distance, not size.
> David Beardsley >>>>

Hi ALL, how could David read my mind? "distance" vs "size" -- that is exactly the source of my confusion.

Please refer to Kolinski [p.3 of http://www.anaphoria.com/kolin.PDF].

In our definition of "Interval", do we imply the distance between the two pitches, or, do we imply its size?

As I said in an earlier message, my problem started with the following data regarding the shruti size. Let us take one-shruti interval as an example:
4 shrutis - 3 shrutis = 1 shruti = 9/8 / 10/9 = 81/80
3 - 2 = 1 shruti = 10/9 / 16/15 = 25/24
2 - 1 shruti = 1 shruti = 16/15 / 91/80 = 256/243
= 16/15 / 25/24 = 128/125
= 135/128 / 81/80 = 25/24
= 135/128 / 256/248 = 32805/32768
(I hope the numbers are OK.)

Now, the differences arise out of intervals -- made up of sizes or distances?

And lastly, how do we define the term interval, arsing out of distance? And define the interval arising out of size? Again, this distinction is very important in understanding 'shruti'.

Regards,
Haresh.

-----Original Message-----
From: Haresh BAKSHI [mailto:hareshbakshi@hotmail.com]

--- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
............

>>>> Monzos def: "The size between two different pitches, or the section
of the linearly-perceived pitch-continuum bounded by those two pitches."
> > I'd say distance, not size.
> David Beardsley >>>>

>Hi ALL, how could David read my mind? "distance" vs "size" -- that is
>exactly the source of my confusion.

All part of my super hero powers. I only enter this realm when I'm
needed. :0

Being a stringed instrument musician, distance makes a whole lot of
sense. Even when I sing, I'm thinking of distance. Hell, if you don't
reduce the distance, size doesn't matter. (I was acting as tour guide in
NYC for some friends from the mid-west yesterday, bear with me).

>And lastly, how do we define the term interval, arsing out of distance?
And >define the interval arising out of size? Again, this distinction is
very >important in understanding 'shruti'.

The distance between notes are best shown on a string...

Here are a few examples:

Play 1/1. Here is the other note: plays 17/16 + 2/1. Some distance
between those notes....and octave plus a 17/16.

Here is 1/1. I sing 5/4. Notice the distance between these notes? Look,
here it is on a stringed instrument.

That is why I like the term distance, I think of the interval on a
string.
These are harmonics on a string, a vertical stack.

An interval is the distance between two pitches or notes, also known as
a dyad.

Space is infinite, while a distance is finite. Even though space is the
place, which is a discussion best saved for another day.

dB

on 11/16/03 2:41 PM, David Beardsley <db@biink.com> wrote:

>
>> I am a little concerned about the term "Interval" -- I will refer to it
>> later.
>
> Monzos def: "The size between two different pitches, or the section of
> the linearly-perceived pitch-continuum bounded by those two pitches."

First thought for suggested edit: "The distance between two pitches, or the
size of the perceived pitch-continuum bounded by those two pitches."

A distance *is* a size. How big or small is the distance? That's a
size--big or small. Also: an inch is a distance; an inch is a size.

"Size" of an interval is a common usage (no?), so no use trying to get rid
of the word, preferring distance instead.

The size of a one-dimensional quantity is the distance between its
endpoints. The pitches are the endpoints. The size of the interval is the
distance between the pitches. There, I think that finishes the thought.
The pitches have a distance (between them). The interval has a size. The
interval is the space between the pitches. The *measurement* of the
interval interval equals the measurement of distance between the pitches.

So if you are going to "split hairs" as someone said, you need to
distinguish a thing from a measurement of the thing. Maybe you want to ask
the question whether the thing exists or only the measurement. However
language does not make this distinction. Language follows its form
regardless of whether all its objects and constructs have an "actual"
referent. So even if an interval has no real existence outside of its
perceived size, in language we have the phrase "size of an interval", and so
in language we have the abstract existence of interval, distinct from its
size.

Insert words like "subjective" where needed but I think they are possibly
redundant. Existence is never explicit. This is my bias in language, which
I think should be recognized as a poetic regardless of technical content.
The desire to make *everything* explicit is a trap because there is always
more. The sense that a description is complete is probably erroneous.
Therefore to me the most complete language is the least embellished. Adding
more begs adding more yet. Adding the least possible creates a true unit of
language - something we could become more sensitive to but have little
experience with. A dictionary that followed this form would be one powerful
book!

I see that I have not read all the threads that I should have read before
replying, but I will leave this as it stands anyway. If people just used
one subject instead of 5 for a given thread, I would not have missed all the
background. ;)

-Kurt

>
> I'd say distance, not size.
>
>
> David Beardsley
>
>
>
>
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hi David and Haresh,

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:

> --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> ............
>
> > >>>> Monzos def: "The size between two different pitches,
> > or the section of the linearly-perceived pitch-continuum
> > bounded by those two pitches."
> > I'd say distance, not size.
> > David Beardsley >>>>
>
> Hi ALL, how could David read my mind? "distance" vs "size"
> -- that is exactly the source of my confusion.
>
> Please refer to Kolinski [p.3 of
> http://www.anaphoria.com/kolin.PDF].
>
> In our definition of "Interval", do we imply the distance
> between the two pitches, or, do we imply its size?

i'd have to say: both. so really, my definition of interval
should have 3 parts: 1) the two pitches themselves, as a
sonic entity, 2) the size of the pitch-space between them,
3) the distance between them in pitch-height.

i'm not really sure why you two see such a distinction
between "size" and "distance" ... in terms of pitch-height,
the two words to me pretty much mean the same thing. i'd
appreciate more clarification.

> As I said in an earlier message, my problem started with
> the following data regarding the shruti size. Let us take
> one-shruti interval as an example:
> 4 shrutis - 3 shrutis = 1 shruti = 9/8 / 10/9 = 81/80
> 3 - 2 = 1 shruti = 10/9 / 16/15 = 25/24
> 2 - 1 shruti = 1 shruti = 16/15 / 91/80 = 256/243
> = 16/15 / 25/24 = 128/125
> = 135/128 / 81/80 = 25/24
> = 135/128 / 256/248 = 32805/32768
> (I hope the numbers are OK.)
>
> Now, the differences arise out of intervals -- made up of sizes
> or distances?
>
> And lastly, how do we define the term interval, arsing out of
> distance? And define the interval arising out of size? Again,
> this distinction is very important in understanding 'shruti'.

ah ... well, now you're getting into distinctions between,
say, absolute difference in pitch-height, for example in cents,
and *scale*-based differences between notes.

for instance, the typical interval names "2nd", "3rd", "4th",
etc., are based on the 7-tone diatonic scale. this is something
which i haven't even yet included in my definition, but it
*is* an important system of pitch/interval measurement in
music ... and since the size of a shruti is so hotly debated
and unclearly understood, i'd say that *that* part of the
definition is an important consideration for you, Haresh.

please, keep the feedback coming. i have to do major
overhauls to both the "interval" and "pitch" definitions.
"scale" could probably use a lot more work too.

-monz

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 11/16/03 2:41 PM, David Beardsley <db@b...> wrote:
> ..............

>>>> A distance *is* a size. How big or small is the distance? That's a > size--big or small. Also: an inch is a distance; an inch is a size.
>>>> "Size" of an interval is a common usage (no?), so no use trying to get rid of the word, preferring distance instead.
>>>> The size of a one-dimensional quantity is the distance between its endpoints. >>>>

Hi Kurt, To quote from one of my earlier postings:
-----------
As I said in an earlier message, my problem started with the following data
regarding the shruti size. Let us take one-shruti interval as an example:
4 shrutis - 3 shrutis = 1 shruti = 9/8 / 10/9 = 81/80
3 - 2 = 1 shruti = 10/9 / 16/15 = 25/24
2 - 1 shruti = 1 shruti = 16/15 / 91/80 = 256/243
= 16/15 / 25/24 = 128/125
= 135/128 / 81/80 = 25/24
= 135/128 / 256/248 = 32805/32768
----------
The point is this: The distance in each case remains one shruti, but the sizes are different (for that same distance).

Of course, this situation may be unique to shruti-s, and, therefore, may not matter substantially in a more general definition of the term Interval.

Regards,
Haresh.

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:
> --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> ............
>
> >>>> Monzos def: "The size between two different pitches, or the
section of the linearly-perceived pitch-continuum bounded by those
two pitches."
> > > I'd say distance, not size.
> > David Beardsley >>>>
>
> Hi ALL, how could David read my mind? "distance" vs "size" -- that
is exactly the source of my confusion.
>
> Please refer to Kolinski [p.3 of
http://www.anaphoria.com/kolin.PDF].

i'm not necessarily in agreement with kolinski, but not sure how
that's relevant.

> In our definition of "Interval", do we imply the distance between
the two pitches, or, do we imply its size?
>
> As I said in an earlier message, my problem started with the
following data regarding the shruti size. Let us take one-shruti
interval as an example:
> 4 shrutis - 3 shrutis = 1 shruti = 9/8 / 10/9 = 81/80
> 3 - 2 = 1 shruti = 10/9 / 16/15 = 25/24
> 2 - 1 shruti = 1 shruti = 16/15 / 91/80 = 256/243
> = 16/15 / 25/24 = 128/125
> = 135/128 / 81/80 = 25/24
> = 135/128 / 256/248 = 32805/32768
> (I hope the numbers are OK.)

yes.

> Now, the differences arise out of intervals -- made up of sizes or
distances?

i'm unsure how to answer your question -- what would the distinction
be?

> And lastly, how do we define the term interval, arsing out of
>distance? And define the interval arising out of size?

perhaps there is no distinction at all, other than better vs. worse
use of the english language.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Insert words like "subjective" where needed but I think they are
>possibly
> redundant.

in some cases, it's important -- pitch is subjective, as haresh
pointed out, depending on so many particular factors including the
details of an individual ear (i really mean ear, as a single person's
left and right ears may perceive pitch differently), while frequency
is objectively measurable. pitch is *primarily* a function of
frequency, but not entirely.

actually even that's too generous, since it's really only true for
sine waves. for more complex timbres, our perception of pitch is
influenced, or sometimes entirely formed, by the upper partial
frequencies, which are not themselves heard as individual pitches,
but from which the brain tries to discern a single "fundamental"
pitch. no definition of pitch will really have much relevance to
objectively measurable facts without mentioning such considerations,
i'm afraid.

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:
> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> > on 11/16/03 2:41 PM, David Beardsley <db@b...> wrote:
> > ..............
>
> >>>> A distance *is* a size. How big or small is the distance?
That's a > size--big or small. Also: an inch is a distance; an inch
is a size.
> >>>> "Size" of an interval is a common usage (no?), so no use
trying to get rid of the word, preferring distance instead.
> >>>> The size of a one-dimensional quantity is the distance between
its endpoints. >>>>
>
> Hi Kurt, To quote from one of my earlier postings:
> -----------
> As I said in an earlier message, my problem started with the
following data
> regarding the shruti size. Let us take one-shruti interval as an
example:
> 4 shrutis - 3 shrutis = 1 shruti = 9/8 / 10/9 = 81/80
> 3 - 2 = 1 shruti = 10/9 / 16/15 = 25/24
> 2 - 1 shruti = 1 shruti = 16/15 / 91/80 = 256/243
> = 16/15 / 25/24 = 128/125
> = 135/128 / 81/80 = 25/24
> = 135/128 / 256/248 = 32805/32768
> ----------
> The point is this: The distance in each case remains one shruti,
>but the sizes are different (for that same distance).

one could just as well say "The size in each case remains one shruti,
but the distances are different (for that same size)."

so i don't think distance vs. size is the appropriate way to make
this distinction. on tuning-math, we might talk about an epimorphism
between the just 5-limit lattice and a 22-tone system (in which case
one could not use 64/45 (610 cents above 1/1) for shruti #12, but
instead 729/512 (612 cents above 1/1) or a subsitute -- see
http://sonic-arts.org/td/erlich/srutipblock.htm), and then one could
discuss the set of just intervals which mapped to 1 degree of the
scale -- this set would include all the ratios you equate with one
shruti above, and more. if you had included only the 1-shruti
intervals that actually occur as steps in the 22-shruti scale (unlike
128:125 and 32805:32768), we could revert to common "tuning" talk and
say that the *generic* interval of 1 step in the shruti scale occurs
in several *specific* sizes. this kind of talk is important in
discussing all kinds of scales, not just the shruti scale.

on 11/17/03 2:14 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> Insert words like "subjective" where needed but I think they are
>> possibly
>> redundant.
>
> in some cases, it's important -- pitch is subjective, as haresh
> pointed out,

Yes, but saying "perceived pitch" or "subjective pitch" all over the place
(for example) is still redundant. Pitch is never anything but subjective,
right? It has no other existence, I don't think.

Rather the definition of pitch should use the word "subjective" and in a
hyper-linked context all occurrences of "pitch" can be linked to that
definition. Following that principle keeps definitions from overloading the
brain with words when read.

> depending on so many particular factors including the
> details of an individual ear (i really mean ear, as a single person's
> left and right ears may perceive pitch differently), while frequency
> is objectively measurable. pitch is *primarily* a function of
> frequency, but not entirely.

Yes, and of course amplitude also influences pitch.

There are *apparently* at least two different uses/meanings of the word
pitch:

(1) perceived/subjective pitch
(2) "Idealized" (lacking a better word) pitch, something which allows
structural (?) music theory to unfold without being bogged down

So maybe this is why the word "subjective" is needed after all. But if this
is true I'd like to be clearer about it, and I think the Pitch page should
include two distinct definitions, recognizing the distinct usages which
apparently already exist.

So what about this distinction?

-Kurt

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

......................

>>>> so i don't think distance vs. size is the appropriate way to make this distinction. on tuning-math, we might talk about an epimorphism between the just 5-limit lattice and a 22-tone system (in which case one could not use 64/45 (610 cents above 1/1) for shruti #12, but instead 729/512 (612 cents above 1/1) or a subsitute -- see
> http://sonic-arts.org/td/erlich/srutipblock.htm), and then one could discuss the set of just intervals which mapped to 1 degree of the scale -- this set would include all the ratios you equate with one shruti above, and more. if you had included only the 1-shruti
intervals that actually occur as steps in the 22-shruti scale (unlike 128:125 and 32805:32768), we could revert to common "tuning" talk and say that the *generic* interval of 1 step in the shruti scale occurs in several *specific* sizes. this kind of talk is important in discussing all kinds of scales, not just the shruti scale. >>>>

Hi Paul, .... on to tuning-math, for the epimorphism mentioned above, and other points. This will be very important for the study of shruti. Off-list, additionally, please, if I have math-related queries which would be common knowledge for the remaining members.

Thanks and regards,
Haresh.

Hi Haresh,

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:
> Hi Kurt, To quote from one of my earlier postings:
> -----------
> As I said in an earlier message, my problem started with the
following data
> regarding the shruti size. Let us take one-shruti interval as an
example:
> 4 shrutis - 3 shrutis = 1 shruti = 9/8 / 10/9 = 81/80
> 3 - 2 = 1 shruti = 10/9 / 16/15 = 25/24
> 2 - 1 shruti = 1 shruti = 16/15 / 91/80 = 256/243
> = 16/15 / 25/24 = 128/125
> = 135/128 / 81/80 = 25/24
> = 135/128 / 256/248 = 32805/32768
> ----------
> The point is this: The distance in each case remains one shruti,
but the sizes are different (for that same distance).
>
> Of course, this situation may be unique to shruti-s, and,
therefore, may not matter substantially in a more general definition
of the term Interval.
>
> Regards,
> Haresh.

If you want my opinion, I think it is clear we are talking here of
different intervals, which possess however some quality which links
them. In most cases, those groupings of intervals have to do with the
internal structure of a scale, rather than the height (or width) of
the intervals.

As it has already been suggested in one of the posts, we may take the
12-eq scale as a reference.
C-E and C-Eb are two different intervals, that qualify however
as "thirds". Although the difference between those intervals equals
the one found in C-F;C-E, the latter two are qualitative more
distinct, being a comparison between a perfect fourth and a major
third.

No one would of course deny this nomenclature, neither its
usefulness. However, these schemes are based on scale' structure, and
have no direct relation with the measure of the intervals.
So one has a quantitative definition of the interval, and apart from
that, a series of useful labelling that qualify intervals in some way.

To take a funny example, one has even in 12-eq the tritone's example
[sqrt(2)], which is a unique interval but has two different attached
labels (namely, aug 4th or dim 5th).

Max.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > depending on so many particular factors including the
> > details of an individual ear (i really mean ear, as a single
person's
> > left and right ears may perceive pitch differently), while
frequency
> > is objectively measurable. pitch is *primarily* a function of
> > frequency, but not entirely.
>
> Yes, and of course amplitude also influences pitch.

Haresh mentioned that, and I mentioned him, so repeating it would
have been redundant :)

> There are *apparently* at least two different uses/meanings of the
word
> pitch:
>
> (1) perceived/subjective pitch

> (2) "Idealized" (lacking a better word) pitch, something which
allows
> structural (?) music theory to unfold without being bogged down
>
> So maybe this is why the word "subjective" is needed after all.
But if this
> is true I'd like to be clearer about it, and I think the Pitch page
should
> include two distinct definitions, recognizing the distinct usages
which
> apparently already exist.
>
> So what about this distinction?
>
> -Kurt

I think you may have an important point.

http://www.mmk.ei.tum.de/persons/ter/top/defpitch.html

(i've posted it before, and it's very confusing to most people, since
when terhardt speaks of the tone of a musical instrument evoking
multiple pitches at the same time, he's talking, rightly,
about "subharmonics" rather than harmonics -- but i think his pages
are essential reading)

just keep in mind that terhardt is biased against periodicity
theories of pitch, which are currently in greater favor than during
terhardt's heyday.

Hello All,

For pedestrians such as myself, is there anything wrong or incomplete or insufficient or inappropriate to think of musical intervals as being "musical distances" between two musical sounds measured by some simple metric? Cents as a metric on the logarithmic scale is fine with me.

Please ignore this question if it has already been suggested and beaten to death.

Thanks.

Can Akkoc

Haresh BAKSHI <hareshbakshi@hotmail.com> wrote:
--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

......................

>>>> so i don't think distance vs. size is the appropriate way to make this distinction. on tuning-math, we might talk about an epimorphism between the just 5-limit lattice and a 22-tone system (in which case one could not use 64/45 (610 cents above 1/1) for shruti #12, but instead 729/512 (612 cents above 1/1) or a subsitute -- see
> http://sonic-arts.org/td/erlich/srutipblock.htm), and then one could discuss the set of just intervals which mapped to 1 degree of the scale -- this set would include all the ratios you equate with one shruti above, and more. if you had included only the 1-shruti
intervals that actually occur as steps in the 22-shruti scale (unlike 128:125 and 32805:32768), we could revert to common "tuning" talk and say that the *generic* interval of 1 step in the shruti scale occurs in several *specific* sizes. this kind of talk is important in discussing all kinds of scales, not just the shruti scale. >>>>

Hi Paul, .... on to tuning-math, for the epimorphism mentioned above, and other points. This will be very important for the study of shruti. Off-list, additionally, please, if I have math-related queries which would be common knowledge for the remaining members.

Thanks and regards,
Haresh.

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--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
> Hello All,
>
> For pedestrians such as myself, is there anything wrong or
>incomplete or insufficient or inappropriate to think of musical
>intervals as being "musical distances" between two musical sounds
>measured by some simple metric? Cents as a metric on the logarithmic
>scale is fine with me.

Of course there's nothing wrong with that. Haresh, however, was
inquiring about a different, more abstract concept, which has
importance, for example, in Western cognitive music theory and Indian
shruti theory. The musical importance of this concept can be shown
with a simple Western example -- while the minor 3rd and augmented
2nd are the same number of cents in 12-tone equal temperament, they
sound utterly different -- the former functioning as a consonance and
the latter as a dissonance -- in the context of modal or tonal
Western music. To an analyst of Western music prior to 1900, treating
the two intervals the same way would be an egregious error, even if
they always occured with the same measure in cents.

I'm still awaiting further inquiry from Haresh, either on the tuning-
math list or privately . . .

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
....

>>>> I'm still awaiting further inquiry from Haresh, either on the tuning-math list or privately . . .>>>>

Hi Paul, as the discussion on 'pitch' continues, I find myself in the area where what I need to do is, not only 're-thinking', but also re-engineering of some of my concepts on shruti. I am in the process of going through, one more time, the few verses on shruti in Natyashastra and Sangitaratnakaram, for a day or two, before I go back to the query -- tuning-math list or privately.

Regards,
Haresh.

Thanks Paul for clarifying this point. Obviously I need to do more homework.

Could this musical phenomenon be related to the ordering of musical intervals in a sequence of musical intervals - as in a melody line?

Can

Paul Erlich <paul@stretch-music.com> wrote:
--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
> Hello All,
>
> For pedestrians such as myself, is there anything wrong > or incomplete or insufficient or inappropriate to think > of musical intervals as being "musical distances" > between two musical sounds measured by some simple > metric? Cents as a metric on the logarithmic scale is > fine with me.

Of course there's nothing wrong with that. Haresh, however, was inquiring about a different, more abstract concept, which has importance, for example, in Western cognitive music theory and Indian shruti theory. The musical importance of this concept can be shown with a simple Western example -- while the minor 3rd and augmented 2nd are the same number of cents in 12-tone equal temperament, they sound utterly different -- the former functioning as a consonance and the latter as a dissonance -- in the context of modal or tonal Western music. To an analyst of Western music prior to 1900, treating the two intervals the same way would be an egregious error, even if they always occured with the same measure in cents.

I'm still awaiting further inquiry from Haresh, either on the tuning-math list or privately . . .

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hi Can and paul,

> Paul Erlich <paul@s...> wrote:
>
> --- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
> >
> > Hello All,
> >
> > For pedestrians such as myself, is there anything
> > wrong or incomplete or insufficient or inappropriate

really covering the bases there, eh Can? :)

> > to think of musical intervals as being "musical distances"
> > between two musical sounds measured by some simple
> > metric? Cents as a metric on the logarithmic scale is
> > fine with me.
>
> Of course there's nothing wrong with that. Haresh,
> however, was inquiring about a different, more abstract
> concept, hich has importance, for example, in Western
> cognitive music theory and Indian shruti theory. The
> musical importance of this concept can be shown with
> a simple Western example -- while the minor 3rd and
> augmented 2nd are the same number of cents in 12-tone
> equal temperament, they sound utterly different -- the
> former functioning as a consonance and the latter as
> a dissonance -- in the context of modal or tonal Western
> music. To an analyst of Western music prior to 1900,
> treating the two intervals the same way would be an
> egregious error, even if they always occured with the
> same measure in cents.

paul's exactly right.

in fact, IMO, the cases where they are very close or
the same size in cents *are* in some ways the most
interesting, because the tonal key-centric "meaning"
of the chord progessions is given away by the spelling
of the notation, but not in the actual audible frequencies.

(the latter *would* be the case in a non-12edo meantone,
such as 19edo or 31edo, for example.)

--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:

> Thanks Paul for clarifying this point. Obviously I need to
> do more homework.
>
> Could this musical phenomenon be related to the ordering
> of musical intervals in a sequence of musical intervals
> - as in a melody line?
>
> Can

Can, the tonal phenomenon described by paul is attributable
to a wide variety of different causes. pitch is the obvious
one which you started with, and i already mentioned notation
as another.

others could be:

- the types of rhythmic cells used in the piece,

- whether or not there are long sections with drones,

- the ebb-and-flow of the emotional aspect,
and with that -- tempi, volume dynamics, and phrasing,

- literary and/or pictorial "programs" that underly
dramatic works (operas, etc.), and purely instrumental music,

- religious, political, cultural etc. rituals associated with
musical pieces,

- and etc. etc. etc., on and on forever.

exactly what constitutes these kinds of tonal phenomena
have been and are the study of so many music-theorists,
and also of real or self-proclaimed experts in other
fields such as those i mention in the tabulation above,
thruout the world's entire written history. and the
list of publications keeps growing exponenentially...

it's a truly fascinating subject, precisely *because*
it's associated with so many other aspects of life.

(now i'm waxing poetic ... time to stop...)

-monz

--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
> Thanks Paul for clarifying this point. Obviously I need to do more
homework.
>
> Could this musical phenomenon be related to the ordering of musical
>intervals in a sequence of musical intervals - as in a melody line?

Ordering in a melody seems to have little to do with it -- instead
the diatonic scale itself, however ordered or presented musically,
seems to convey a sort of 'measure' to the listener in terms of its
generic intervals (that is, intervals considered as number of steps
in the scale, irrespective of specific or 'objective' size), which in
some respects is preserved even under chromatic alteration of the
diatonic scale degrees (even such as makes the interval 'objectively'
the same size as that of a different generic interval). Much recent
academic music theory by people like Clough, Clampitt, Myerson,
Agmon, Browne, etc., as well as some older theories such as those of
Rothenberg, is concerned with explaining this fact through the
structure of the diatonic scale. It happened to just come up on
tuning-math as well, where we have some less 12-equal-biased and
perhaps mathematically prettier approaches to the question . . .

Joe,

Thanks for a very stimulating response to my primitive questions and comments on tonal phenomena.

I have a sneaky suspicion that your reference "the ebb-and-flow of the emotional aspect, and with that -- tempi, volume dynamics, and phrasing," may very well be what master musicians of traditional Turkish music call "cheshni and TAVIR". Please do not ask me to transliterate these terms. They may be of Turkish, Arabic or Farisi (Persian) descent or a hybrid of those three languages.

Kindest regards,

Can

monz <monz@attglobal.net> wrote:
hi Can and paul,

> Paul Erlich <paul@s...> wrote:
>
> --- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
> >
> > Hello All,
> >
> > For pedestrians such as myself, is there anything
> > wrong or incomplete or insufficient or inappropriate

really covering the bases there, eh Can? :)

> > to think of musical intervals as being "musical distances"
> > between two musical sounds measured by some simple
> > metric? Cents as a metric on the logarithmic scale is
> > fine with me.
>
> Of course there's nothing wrong with that. Haresh,
> however, was inquiring about a different, more abstract
> concept, hich has importance, for example, in Western
> cognitive music theory and Indian shruti theory. The
> musical importance of this concept can be shown with
> a simple Western example -- while the minor 3rd and
> augmented 2nd are the same number of cents in 12-tone
> equal temperament, they sound utterly different -- the
> former functioning as a consonance and the latter as
> a dissonance -- in the context of modal or tonal Western
> music. To an analyst of Western music prior to 1900,
> treating the two intervals the same way would be an
> egregious error, even if they always occured with the
> same measure in cents.

paul's exactly right.

in fact, IMO, the cases where they are very close or
the same size in cents *are* in some ways the most
interesting, because the tonal key-centric "meaning"
of the chord progessions is given away by the spelling
of the notation, but not in the actual audible frequencies.

(the latter *would* be the case in a non-12edo meantone,
such as 19edo or 31edo, for example.)

--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:

> Thanks Paul for clarifying this point. Obviously I need to
> do more homework.
>
> Could this musical phenomenon be related to the ordering
> of musical intervals in a sequence of musical intervals
> - as in a melody line?
>
> Can

Can, the tonal phenomenon described by paul is attributable
to a wide variety of different causes. pitch is the obvious
one which you started with, and i already mentioned notation
as another.

others could be:

- the types of rhythmic cells used in the piece,

- whether or not there are long sections with drones,

- the ebb-and-flow of the emotional aspect,
and with that -- tempi, volume dynamics, and phrasing,

- literary and/or pictorial "programs" that underly
dramatic works (operas, etc.), and purely instrumental music,

- religious, political, cultural etc. rituals associated with
musical pieces,

- and etc. etc. etc., on and on forever.

exactly what constitutes these kinds of tonal phenomena
have been and are the study of so many music-theorists,
and also of real or self-proclaimed experts in other
fields such as those i mention in the tabulation above,
thruout the world's entire written history. and the
list of publications keeps growing exponenentially...

it's a truly fascinating subject, precisely *because*
it's associated with so many other aspects of life.

(now i'm waxing poetic ... time to stop...)

-monz

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Wow ! Thanks Paul. This stuff is way over my head.

However, my gut whispers to me that this tonal phenoema might have something to do with the "seyir" issue in Turkish music. I will keep digging in.

Can

Paul Erlich <paul@stretch-music.com> wrote:
--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
> Thanks Paul for clarifying this point. Obviously I need to do more
homework.
>
> Could this musical phenomenon be related to the ordering of musical
>intervals in a sequence of musical intervals - as in a melody line?

Ordering in a melody seems to have little to do with it -- instead
the diatonic scale itself, however ordered or presented musically,
seems to convey a sort of 'measure' to the listener in terms of its
generic intervals (that is, intervals considered as number of steps
in the scale, irrespective of specific or 'objective' size), which in
some respects is preserved even under chromatic alteration of the
diatonic scale degrees (even such as makes the interval 'objectively'
the same size as that of a different generic interval). Much recent
academic music theory by people like Clough, Clampitt, Myerson,
Agmon, Browne, etc., as well as some older theories such as those of
Rothenberg, is concerned with explaining this fact through the
structure of the diatonic scale. It happened to just come up on
tuning-math as well, where we have some less 12-equal-biased and
perhaps mathematically prettier approaches to the question . . .

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--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:

> Wow ! Thanks Paul. This stuff is way over my head.

Well, the fact that an augmented 2nd sounds like a dissonance,
completely different than the 'acoustically identical' minor 3rd
which sounds like a consonance, was demonstrated by my high school
music teacher, and shouldn't be too difficult for anyone familiar
with western music to observe. Just try some very simple, short, not
necessarily musical examples in the harmonic minor scale. Then land
on one of these intervals, or use one of them in a melody, and
compare with the result of similarly landing on or using the other.
To me, at least, the difference is astonishingly vivid, and virtually
all Western music theorists seem to agree.

Good to see you actively engaged, Can!

--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:

................

>>>> For pedestrians such as myself, is there anything wrong or incomplete or insufficient or inappropriate to think of musical intervals as being "musical distances" between two musical sounds measured by some simple metric? Cents as a metric on the logarithmic scale is fine with me. >>>>

Hello Dr. Akkoc, thanks for the response. As you have rightly pointed out, the term 'interval', simply, is the distance between things; in music, it is the distance between two sounds/pitches/tones/notes [I prefer 'pitches'] The term 'distance' can be elucidated under Remarks/explanation/note, and can be so many things. No problem.

The complexities start as soon as we see 'interval' in a larger perspective: Like, for example, it means "Distance and relationship between two pitches". And, please look at this: "A set containing all points (or all real numbers) between two given endpoints (the pitches, in the present context)." In addition, the interval can have an assigned name ("perfect fifth"), or an ordinal number (as Monz has pointed out in his dictionary) (the 3rd, for instance). Further, we have, as intervals, the commas and the Schisma, and the Diesis.

Moreover, in the light of the present discussions, I am trying to understand the term 'Ratio'
[See http://sonic-arts.org/dict/ratio.htm].

Lastly, many cultures around the world who do not use the tempered scale have their own names for intervals found in their music. "Shruti" is one such instance. I dare to suspect (in case of the Indian psyche, you can only venture and dare and respectfully submit) that the shruti concept is best represented as:

FROM audible continuum of an infinite number of pitches:
___________________________________________

TO shruti-s
...........................................

TO swara-s

...................... [OR start on the first shruti.]
^ ^ ^ ^ ^ ^ ^

Now, EITHER the swara is a single NUMBER, located as shown by the upward-pointing arrows, each swara with its individual name; OR the swara is a band of shruti-s, *shining* (sounding) as a swara only at the end of the particular band.

Hence my perplexity regarding treating shruti as an individual pitch, or a band of pitches. As I continue reading more responses on the list, and with Paul, I think the picture will become clearer.

Regards,
Haresh.

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:

> Moreover, in the light of the present discussions, I am trying to
>understand the term 'Ratio'
> [See http://sonic-arts.org/dict/ratio.htm].

I wouldn't get too hung up on Partch's pretentious-sounding
definition. If I have 6 apples and you have 4 apples, the ratio of
the number of apples I have to the number of apples you have is 3:2.
Probably this is a familiar concept to you and to most people.

What is the context in the present discussions in which the use of
the term 'ratio' is confusing you?

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...> wrote:
> --- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
>
> ................
> Lastly, many cultures around the world who do not use the tempered scale have their own names for intervals found in their music. "Shruti" is one such instance. I dare to suspect (in case of the Indian psyche, you can only venture and dare and respectfully submit) that the shruti concept is best represented as:
>
> FROM audible continuum of an infinite number of pitches:
> ___________________________________________
>
> TO shruti-s
> ...........................................
>
> TO swara-s
>
> ...................... [OR start on the first shruti.]
> ^ ^ ^ ^ ^ ^ ^
>>>>

The arrows above, completely misaligned on posting the message, should fall according to 4 3 2 4 4 3 2 respectively.

Thanks,
Haresh.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
> wrote:
>
> > Moreover, in the light of the present discussions, I am trying to
> >understand the term 'Ratio'
> > [See http://sonic-arts.org/dict/ratio.htm].
>
> I wouldn't get too hung up on Partch's pretentious-sounding
> definition. If I have 6 apples and you have 4 apples, the ratio of
> the number of apples I have to the number of apples you have is 3:2.
> Probably this is a familiar concept to you and to most people.
>
> What is the context in the present discussions in which the use of
> the term 'ratio' is confusing you? >>>>>>>>

Hi Paul, I quote:

--------------
ratio

a relationship, or interval, expressing the vibrations per second, or cycles, of the two tones concerned, generally in the lowest possible [integer] terms;...simultaneously a representative of a tone and an implicit relationship to a "keynote" -- or unity.
---------------

Is not any interval necessarily also a ratio? And there will be several ways of expressing it (perhaps not only as the vibrations per second, or cycles.) And, now, we seem to prefer to use 'tones' rather than 'pitches' -- any reason? The concept of shruti is already confusing enough, without any additional conundrums. Of course, I quite realize that the limitation of understanding what may be a perfectly simple statement comes from my side.

Regards,
Haresh.

--- In tuning@yahoogroups.com, "Haresh BAKSHI" <hareshbakshi@h...>
wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, "Haresh BAKSHI"
<hareshbakshi@h...>
> > wrote:
> >
> > > Moreover, in the light of the present discussions, I am trying
to
> > >understand the term 'Ratio'
> > > [See http://sonic-arts.org/dict/ratio.htm].
> >
> > I wouldn't get too hung up on Partch's pretentious-sounding
> > definition. If I have 6 apples and you have 4 apples, the ratio
of
> > the number of apples I have to the number of apples you have is
3:2.
> > Probably this is a familiar concept to you and to most people.
> >
> > What is the context in the present discussions in which the use
of
> > the term 'ratio' is confusing you? >>>>>>>>
>
> Hi Paul, I quote:
>
> --------------
> ratio
>
> a relationship, or interval, expressing the vibrations per
second, or cycles, of the two tones concerned, generally in the
lowest possible [integer] terms;...simultaneously a representative of
a tone and an implicit relationship to a "keynote" -- or unity.
> ---------------

That's the pretentious definition i was hoping you wouldn't get to
hung up on, above.

> Is not any interval necessarily also a ratio?

Yes (if it's a specific, not generic, interval), although if the
ratio is an irrational number, it's impossible to express as the
quotient of two integers.

>And there will be several ways of expressing it (perhaps not only as
>the vibrations per second, or cycles.)

Ratios themselves are never expressed as vibrations per second.
ratios are dimensionless quantities, because two quantities with the
same dimension (either vibrations per second, cycles, or whatever)
are divided one by the other, canceling the units. It is true that if
the ratio between the vibrations per second, or however measured
frequencies, is a/b, then the ratio between the periods, or sonic
wavelengths, or idealized string lengths, is b/a. But that's it --
you'll either see the frequency ratio or its reciprocal, never any
other value, for the ratio representing a given specific interval.

>And, now, we seem to prefer to use 'tones' rather than 'pitches' --
>any reason?

If I'm extremely generous to Partch and assume he somehow presciently
foresaw this whole discussion with the terms we've happened to use --
a pretty absurd assumption -- I'd say that 'tones' helps emphasize
the *objective* nature of the phenomena whose vibrations per second,
or cycles, are being divided one by the other.

Perhaps the worst part of Partch's definition is that he doesn't
specify that *division* is used to come up with the ratio, but simply
defines it as 'a relationship, or interval'! Like most of Partch's
definitions, fairly useless, and one has to glean the meaning of his
terms from their context in his book.

Haresh BAKSHI <hareshbakshi@hotmail.com> wrote:
--- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:

................

>>>> For pedestrians such as myself, is there anything wrong or incomplete or insufficient or inappropriate to think of musical intervals as being "musical distances" between two musical sounds measured by some simple metric? Cents as a metric on the logarithmic scale is fine with me. >>>>

Hello Dr. Akkoc, thanks for the response. As you have rightly pointed out, the term 'interval', simply, is the distance between things; in music, it is the distance between two sounds/pitches/tones/notes [I prefer 'pitches'] The term 'distance' can be elucidated under Remarks/explanation/note, and can be so many things. No problem.

The complexities start as soon as we see 'interval' in a larger perspective: Like, for example, it means "Distance and relationship between two pitches". And, please look at this: "A set containing all points (or all real numbers) between two given endpoints (the pitches, in the present context)." In addition, the interval can have an assigned name ("perfect fifth"), or an ordinal number (as Monz has pointed out in his dictionary) (the 3rd, for instance). Further, we have, as intervals, the commas and the Schisma, and the Diesis.

Moreover, in the light of the present discussions, I am trying to understand the term 'Ratio'
[See http://sonic-arts.org/dict/ratio.htm].

Lastly, many cultures around the world who do not use the tempered scale have their own names for intervals found in their music. "Shruti" is one such instance. I dare to suspect (in case of the Indian psyche, you can only venture and dare and respectfully submit) that the shruti concept is best represented as:

FROM audible continuum of an infinite number of pitches:
___________________________________________

TO shruti-s
...........................................

TO swara-s

...................... [OR start on the first shruti.]
^ ^ ^ ^ ^ ^ ^

Now, EITHER the swara is a single NUMBER, located as shown by the upward-pointing arrows, each swara with its individual name; OR the swara is a band of shruti-s, *shining* (sounding) as a swara only at the end of the particular band.

Hence my perplexity regarding treating shruti as an individual pitch, or a band of pitches. As I continue reading more responses on the list, and with Paul, I think the picture will become clearer.

Regards,
Haresh.

-------------------

Hello Haresh and All,

Thank you for taking the time to respond to my questions.

It looks like I found myself unintentionally in the middle of a hot exchange that, I must admit, I had not been following very closely.

If you would allow me, I will stand in the sidelines for the time being and not post until such time when things begin to clear from my perspective.

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Hi All,
of course there is a difference between augmented maj. second and
minor third - this belongs to our conception of joining harmonies in listeing.
I mean, we join some notes to a harmonic structure.
Play e.g. the fifth CG on the piano. Add an e after some seconds.
Add an b after some more seconds. No more CG, but g#' and d#".
No more e b, but c''' g'''. Then go fast the whole chord downwards. Now: do the C G and the c''' g''' sound as if in tune ?
This is enharmonic change, and we all know it well.
And, go one step further with Alois H�ba: "What is it that links two tones together, even when they are separated by SILENCE ?"
Our ears are schooled by "tonal" music, and this is what disorientates us in the first moment when we encounter dodekaphonic works.
So we got to make a difference between the strict concept of 12 tet (each interval is so and so many cents large, independent of it's musical FUNCTION), and the PERCEPTION in musical context.
So far 4 today
Bye
Werner

on 11/18/03 1:43 PM, Paul Erlich <paul@stretch-music.com> wrote:

> http://www.mmk.ei.tum.de/persons/ter/top/defpitch.html
>
> (i've posted it before, and it's very confusing to most people,

Well it didn't seem confusing without your elucidation... ;)

> since
> when terhardt speaks of the tone of a musical instrument evoking
> multiple pitches at the same time, he's talking, rightly,
> about "subharmonics" rather than harmonics

Well it seems to me on this page:

http://www.mmk.ei.tum.de/persons/ter/top/pitch.html

since he refers to "the strings of the piano, guitar, etc., and bells" it
seems that he is talking about simple pitch notions breaking down when the
partials are not harmonic.

However this page:

http://www.mmk.ei.tum.de/persons/ter/top/virtualp.html

hints at what I suppose you might mean by reference to subharmonics - though
it didn't seem to be quite what you were saying. I am used to "subharmonic"
being used when referring to a single pitch for which multiple subharmonics
are present, such as is done "artificially" in utonal constructs. Rather it
sounds like he and (maybe) you are talking about multiple harmonics of an
implied fundamental being present with the fundamental implied by virtue of
being subharmonic to each of the harmonics present.

This makes it seem a little confusing when the word "subharmonic" is quoted
out of context here. To me he is still talking about harmonics, it is just
that the (possibly absent) fundamental is simultaneously subharmonic to the
various harmonics present.

> -- but i think his pages
> are essential reading)
>
> just keep in mind that terhardt is biased against periodicity
> theories of pitch, which are currently in greater favor than during
> terhardt's heyday.

In the main page you reference it looked like he was against confounding
perceptual periodicity of pitch with "height" of pitch, into one combined
concept of pitch. This seems reasonable.

Regarding his rejection of a time-domain model, I wonder whether he was not
considering neural models. It strikes me that with sufficient nonlinearity
(as with neural firing) that time-domain models could account well for the
implied fundamental being actually present as a *generated* signal, even in
spite of individual harmonics being presented to individual ears.

-Kurt

on 11/19/03 9:58 AM, Haresh BAKSHI <hareshbakshi@hotmail.com> wrote:

> --- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
>
> ................
>
>>>>> For pedestrians such as myself, is there anything wrong or incomplete or
>>>>> insufficient or inappropriate to think of musical intervals as being
>>>>> "musical distances" between two musical sounds measured by some simple
>>>>> metric? Cents as a metric on the logarithmic scale is fine with me. >>>>
>
> Hello Dr. Akkoc, thanks for the response. As you have rightly pointed out, the
> term 'interval', simply, is the distance between things; in music, it is the
> distance between two sounds/pitches/tones/notes [I prefer 'pitches'] The term
> 'distance' can be elucidated under Remarks/explanation/note, and can be so
> many things. No problem.
>
> The complexities start as soon as we see 'interval' in a larger perspective:
> Like, for example, it means "Distance and relationship between two pitches".
> And, please look at this: "A set containing all points (or all real numbers)
> between two given endpoints (the pitches, in the present context)."

Yes, well in spite of the fact that you said "real numbers" implying a
continuous range of pitches, I couldn't help but reading into what you said
that a discrete set of pitches (e.g. a scale) might be assumed in some
contexts, and this makes the definition of the "size" of an interval get
even more interesting.

Alas, I should know better than to hope for orthogonality in language (or in
a dictionary). I suppose there is a contant (elitist?) desire to refine,
and a larger and important cultural tendency to inter-relate words and
meanings in always new ways, through the actual use of language in relation
to shared experience.

-Kurt

> In
> addition, the interval can have an assigned name ("perfect fifth"), or an
> ordinal number (as Monz has pointed out in his dictionary) (the 3rd, for
> instance). Further, we have, as intervals, the commas and the Schisma, and the
> Diesis.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 11/18/03 1:43 PM, Paul Erlich <paul@s...> wrote:
>
> > http://www.mmk.ei.tum.de/persons/ter/top/defpitch.html
> >
> > (i've posted it before, and it's very confusing to most people,
>
> Well it didn't seem confusing without your elucidation... ;)

Oops! And thanks for paying attention -- hopefully Monz is too!

> > since
> > when terhardt speaks of the tone of a musical instrument evoking
> > multiple pitches at the same time, he's talking, rightly,
> > about "subharmonics" rather than harmonics
>
> Well it seems to me on this page:
>
> http://www.mmk.ei.tum.de/persons/ter/top/pitch.html
>
> since he refers to "the strings of the piano, guitar, etc., and
bells" it
> seems that he is talking about simple pitch notions breaking down
when the
> partials are not harmonic.

Simple periodicity pitch notions breaking down, correct.

> However this page:
>
> http://www.mmk.ei.tum.de/persons/ter/top/virtualp.html
>
> hints at what I suppose you might mean by reference to
subharmonics - though
> it didn't seem to be quite what you were saying. I am used
to "subharmonic"
> being used when referring to a single pitch for which multiple
subharmonics
> are present, such as is done "artificially" in utonal constructs.

Yes, if that's what I meant, I would have simply written subharmonics
without any quotation marks around it.

> Rather it
> sounds like he and (maybe) you are talking about multiple harmonics
of an
> implied fundamental being present with the fundamental implied by
virtue of
> being subharmonic to each of the harmonics present.

Yes, and also the possibility of more than one implied fundamental
being evoked by a given set of "putatively harmonic" partials.

> This makes it seem a little confusing when the word "subharmonic"
is quoted
> out of context here.

Sorry, I put it in quotes not because I was quoting it, but because I
meant to avoid implying (as surely you already see) that any
subharmonic series or utonal chord is actually physically present.

> To me he is still talking about harmonics, it is just
> that the (possibly absent) fundamental is simultaneously
subharmonic to the
> various harmonics present.

Well, the main point is that when he's talking about *multiple
pitches*, he's *not* talking about multiple harmonics or multiple
partials, rather he's talking about multiple putative fundamentals,
all of which are "subharmonic" to some degree to the various partials
present. That's what I think was very confusing to readers of this
page in previous discussions on these lists.

> > -- but i think his pages
> > are essential reading)
> >
> > just keep in mind that terhardt is biased against periodicity
> > theories of pitch, which are currently in greater favor than
during
> > terhardt's heyday.
>
> In the main page you reference it looked like he was against
confounding
> perceptual periodicity of pitch with "height" of pitch, into one
combined
> concept of pitch. This seems reasonable.
>
> Regarding his rejection of a time-domain model, I wonder whether he
was not
> considering neural models. It strikes me that with sufficient
nonlinearity
> (as with neural firing) that time-domain models could account well
for the
> implied fundamental being actually present as a *generated* signal,
even in
> spite of individual harmonics being presented to individual ears.
>
> -Kurt

I'd be interested in hearing more about your ideas. Here's a recent
piece of research from someone less "ossified" than Terhardt which
might get your juices flowing:

http://homepage.mac.com/cariani/CarianiWebsite/TramoHarmony.pdf

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
>
> > Wow ! Thanks Paul. This stuff is way over my head.
>
> Well, the fact that an augmented 2nd sounds like a dissonance,
> completely different than the 'acoustically identical' minor 3rd
> which sounds like a consonance, was demonstrated by my high school
> music teacher, and shouldn't be too difficult for anyone familiar
> with western music to observe. Just try some very simple, short,
not
> necessarily musical examples in the harmonic minor scale. Then land
> on one of these intervals, or use one of them in a melody, and
> compare with the result of similarly landing on or using the other.
> To me, at least, the difference is astonishingly vivid, and
virtually
> all Western music theorists seem to agree.
>

Paul, could you give a specific example of how an augmented 2nd seems
different from a minor 3rd?

thanks,
Jeff

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
> >
> > > Wow ! Thanks Paul. This stuff is way over my head.
> >
> > Well, the fact that an augmented 2nd sounds like a dissonance,
> > completely different than the 'acoustically identical' minor 3rd
> > which sounds like a consonance, was demonstrated by my high
school
> > music teacher, and shouldn't be too difficult for anyone familiar
> > with western music to observe. Just try some very simple, short,
> not
> > necessarily musical examples in the harmonic minor scale. Then
land
> > on one of these intervals, or use one of them in a melody, and
> > compare with the result of similarly landing on or using the
other.
> > To me, at least, the difference is astonishingly vivid, and
> virtually
> > all Western music theorists seem to agree.
> >
>
> Paul, could you give a specific example of how an augmented 2nd
seems
> different from a minor 3rd?
>
> thanks,
> Jeff

Well, hopefully you'll play around yourself, as suggested above. If
you use the C harmonic minor scale, the augmented second is Ab-B,
while the minor thirds include C-Eb, F-Ab, and B-D. If you still need
something more specific, let me know. And I hope George Secor chimes
in, since he was just discussing this difference on the tuning-math
list.

Anyway, this has little to do with my objections to your webpage, so
feel free to ignore it (or me) for now . . .

-Paul

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
> > --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > > --- In tuning@yahoogroups.com, Can Akkoc <can193849@y...> wrote:
> > >
> > > > Wow ! Thanks Paul. This stuff is way over my head.
> > >
> > > Well, the fact that an augmented 2nd sounds like a dissonance,
> > > completely different than the 'acoustically identical' minor
3rd
> > > which sounds like a consonance, was demonstrated by my high
school
> > > music teacher, and shouldn't be too difficult for anyone
familiar
> > > with western music to observe. Just try some very simple,
short, not
> > > necessarily musical examples in the harmonic minor scale. Then
land
> > > on one of these intervals, or use one of them in a melody, and
> > > compare with the result of similarly landing on or using the
other.
> > > To me, at least, the difference is astonishingly vivid, and
virtually
> > > all Western music theorists seem to agree.
> >
> > Paul, could you give a specific example of how an augmented 2nd
seems
> > different from a minor 3rd?
> >
> > thanks,
> > Jeff
>
> Well, hopefully you'll play around yourself, as suggested above. If
> you use the C harmonic minor scale, the augmented second is Ab-B,
> while the minor thirds include C-Eb, F-Ab, and B-D. If you still
need
> something more specific, let me know. And I hope George Secor
chimes
> in, since he was just discussing this difference on the tuning-math
> list.
> ...
> -Paul

Jeff,

For a proper understanding of how intervals function in the diatonic
system, it is helpful to recognize that the diatonic system not only
*can* exist outside 12-ET, but that historically it *predates* 12-
ET. Diatonic intervals are based on a sequence of tones in an open
chain of fifths:

... –6 –5 –4 –3 –2 –1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 ...
... Gb Db Ab Eb Bb F_ C_ G_ D_ A_ E_ B_ F# C# G# D# ...
... d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7 A4 A1 A5 A2 ...

The first line labels the number of positions in the chain, counting
from C. The second line gives the note names (with the underscore
characters are standing in for natural signs), and the third line
gives the interval the tone makes with C: d = diminished, m = minor,
P = perfect, M = major, A = augmented.

In the C harmonic minor scale:
C D Eb F G Ab B C
the augmented 2nd occurs between Ab and B (two consecutive tones in
the scale), whereas each of the minor 3rds, such as C-Eb, have
another scale tone occurring between. Notice that in the chain of
fifths above the upper tone of a minor 3rd is 3 positions to the left
of the lower tone, but for an augmented 2nd the upper tone will be 9
positions to the right. In historical tunings (such as the meantone
temperament) in which the fifths were different in size from those in
12-ET, the enharmonic sharps and flats are different pitches, so
intervals such as the minor 3rd and augmented 2nd are different in
size. Since the fifths of 12-ET are exactly 7/12 of an octave, a
chain of 12 fifths (reduced by 7 octaves) will return you to the
starting tone, which causes the 12-ET pitches that represent
enharmonically related tones to be exactly the same. This also
causes enharmonically related intervals in 12-ET to be the same size;
however, they actually have *very different functions* in diatonic
scales (as demonstrated by the chain-of-fifths diagram above) that we
are able to hear in a diatonic context.

As Paul suggested, you should play around with a harmonic minor scale
to hear this for yourself. And if you get a chance, a real ear-
opener is to listen to something played in meantone temperament (in a
minor key) that contains augmented 2nds or augmented 5ths, which
sound very different from the minor 3rds and 6ths to which they are
enharmonically related.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> Jeff,
>
> For a proper understanding of how intervals function in the
diatonic
> system, it is helpful to recognize that the diatonic system not
only
> *can* exist outside 12-ET, but that historically it *predates* 12-
> ET. Diatonic intervals are based on a sequence of tones in an open
> chain of fifths:
>
> ... –6 –5 –4 –3 –2 –1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 ...
> ... Gb Db Ab Eb Bb F_ C_ G_ D_ A_ E_ B_ F# C# G# D# ...
> ... d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7 A4 A1 A5 A2 ...
>
> The first line labels the number of positions in the chain,
counting
> from C. The second line gives the note names (with the underscore
> characters are standing in for natural signs), and the third line
> gives the interval the tone makes with C: d = diminished, m =
minor,
> P = perfect, M = major, A = augmented.
>
> In the C harmonic minor scale:
> C D Eb F G Ab B C
> the augmented 2nd occurs between Ab and B (two consecutive tones in
> the scale), whereas each of the minor 3rds, such as C-Eb, have
> another scale tone occurring between. Notice that in the chain of
> fifths above the upper tone of a minor 3rd is 3 positions to the
left
> of the lower tone, but for an augmented 2nd the upper tone will be
9
> positions to the right. In historical tunings (such as the
meantone
> temperament) in which the fifths were different in size from those
in
> 12-ET, the enharmonic sharps and flats are different pitches, so
> intervals such as the minor 3rd and augmented 2nd are different in
> size. Since the fifths of 12-ET are exactly 7/12 of an octave, a
> chain of 12 fifths (reduced by 7 octaves) will return you to the
> starting tone, which causes the 12-ET pitches that represent
> enharmonically related tones to be exactly the same. This also
> causes enharmonically related intervals in 12-ET to be the same
size;
> however, they actually have *very different functions* in diatonic
> scales (as demonstrated by the chain-of-fifths diagram above) that
we
> are able to hear in a diatonic context.

While George is absolutely correct that the diatonic system,
historically, was almost always tuned as a chain of fifths, one can
base the explanation instead on the 'fiction' of the just intonation
diatonic system, which is not a chain of fifths. One then sees that
the argument can be detached from the particular interval of the
fifth, and framed in more general terms as one of 'periodicity'.
Hopefully this paper (skip the first section, marked 'optional', if
you wish) will give you the ideas necessary to do this:

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

> As Paul suggested, you should play around with a harmonic minor
scale
> to hear this for yourself. And if you get a chance, a real ear-
> opener is to listen to something played in meantone temperament (in
a
> minor key) that contains augmented 2nds or augmented 5ths, which
> sound very different from the minor 3rds and 6ths to which they are
> enharmonically related.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
[snip]

Paul and George,

thanks for your replies. I am traveling this week, so
it may be a few days before I can process the information.

--Jeff

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
> wrote:
> > > --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > > > --- In tuning@yahoogroups.com, Can Akkoc <can193849@y...>
wrote:
> > > >
> > > > > Wow ! Thanks Paul. This stuff is way over my head.
> > > >
> > > > Well, the fact that an augmented 2nd sounds like a
dissonance,
> > > > completely different than the 'acoustically identical' minor
> 3rd
> > > > which sounds like a consonance, was demonstrated by my high
> school
> > > > music teacher, and shouldn't be too difficult for anyone
> familiar
> > > > with western music to observe. Just try some very simple,
> short, not
> > > > necessarily musical examples in the harmonic minor scale.
Then
> land
> > > > on one of these intervals, or use one of them in a melody,
and
> > > > compare with the result of similarly landing on or using the
> other.
> > > > To me, at least, the difference is astonishingly vivid, and
> virtually
> > > > all Western music theorists seem to agree.
> > >
> > > Paul, could you give a specific example of how an augmented 2nd
> seems
> > > different from a minor 3rd?
> > >
> > > thanks,
> > > Jeff
> >
> > Well, hopefully you'll play around yourself, as suggested above.
If
> > you use the C harmonic minor scale, the augmented second is Ab-B,
> > while the minor thirds include C-Eb, F-Ab, and B-D. If you still
> need
> > something more specific, let me know. And I hope George Secor
> chimes
> > in, since he was just discussing this difference on the tuning-
math
> > list.
> > ...
> > -Paul
>
> Jeff,
>
> For a proper understanding of how intervals function in the
diatonic
> system, it is helpful to recognize that the diatonic system not
only
> *can* exist outside 12-ET, but that historically it *predates* 12-
> ET. Diatonic intervals are based on a sequence of tones in an open
> chain of fifths:
>
> ... –6 –5 –4 –3 –2 –1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 ...
> ... Gb Db Ab Eb Bb F_ C_ G_ D_ A_ E_ B_ F# C# G# D# ...
> ... d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7 A4 A1 A5 A2 ...
>
> The first line labels the number of positions in the chain,
counting
> from C. The second line gives the note names (with the underscore
> characters are standing in for natural signs), and the third line
> gives the interval the tone makes with C: d = diminished, m =
minor,
> P = perfect, M = major, A = augmented.
>
> In the C harmonic minor scale:
> C D Eb F G Ab B C
> the augmented 2nd occurs between Ab and B (two consecutive tones in
> the scale), whereas each of the minor 3rds, such as C-Eb, have
> another scale tone occurring between. Notice that in the chain of
> fifths above the upper tone of a minor 3rd is 3 positions to the
left
> of the lower tone, but for an augmented 2nd the upper tone will be
9
> positions to the right.

Ok, this confuses me. In the standard circle of 5ths, they are
all 3 steps counter-clockwise ( Ab --> B or C --> Eb )

In historical tunings (such as the meantone
> temperament) in which the fifths were different in size from those
in
> 12-ET, the enharmonic sharps and flats are different pitches, so
> intervals such as the minor 3rd and augmented 2nd are different in
> size. Since the fifths of 12-ET are exactly 7/12 of an octave, a
> chain of 12 fifths (reduced by 7 octaves) will return you to the
> starting tone, which causes the 12-ET pitches that represent
> enharmonically related tones to be exactly the same. This also
> causes enharmonically related intervals in 12-ET to be the same
size;
> however, they actually have *very different functions* in diatonic
> scales (as demonstrated by the chain-of-fifths diagram above) that
we
> are able to hear in a diatonic context.
>
> As Paul suggested, you should play around with a harmonic minor
scale
> to hear this for yourself.

I have tried, but I don't find anything on my 12-et piano...
probably because I don't know how to create the musical context
that would produce the effect that you both are describing.
Where is Paul's high school music teacher when we need her? :)

--Jeff

And if you get a chance, a real ear-
> opener is to listen to something played in meantone temperament (in
a
> minor key) that contains augmented 2nds or augmented 5ths, which
> sound very different from the minor 3rds and 6ths to which they are
> enharmonically related.
>
> --George

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> > Jeff,
> > ...
> > For a proper understanding of how intervals function in the
diatonic
> > system, it is helpful to recognize that the diatonic system not
only
> > *can* exist outside 12-ET, but that historically it *predates* 12-
> > ET. Diatonic intervals are based on a sequence of tones in an
open
> > chain of fifths:
> >
> > ... –6 –5 –4 –3 –2 –1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 ...
> > ... Gb Db Ab Eb Bb F_ C_ G_ D_ A_ E_ B_ F# C# G# D# ...
> > ... d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7 A4 A1 A5 A2 ...
> >
> > The first line labels the number of positions in the chain,
counting
> > from C. The second line gives the note names (with the
underscore
> > characters are standing in for natural signs), and the third line
> > gives the interval the tone makes with C: d = diminished, m =
minor,
> > P = perfect, M = major, A = augmented.
> >
> > In the C harmonic minor scale:
> > C D Eb F G Ab B C
> > the augmented 2nd occurs between Ab and B (two consecutive tones
in
> > the scale), whereas each of the minor 3rds, such as C-Eb, have
> > another scale tone occurring between. Notice that in the chain
of
> > fifths above the upper tone of a minor 3rd is 3 positions to the
left
> > of the lower tone, but for an augmented 2nd the upper tone will
be 9
> > positions to the right.
>
> Ok, this confuses me. In the standard circle of 5ths, they are
> all 3 steps counter-clockwise ( Ab --> B or C --> Eb )

This is true only in 12-ET. But in 12-ET it is still true that Ab --
> B is 9 steps clockwise. The 5ths are in a circle only in the
special case of 12-ET (assuming that all 5ths are the same size).
The principles of traditional harmony developed at a time when 12-ET
was not the predominant tuning, and as a result its underlying theory
(and practice) allows that enharmonically related sharps and flats
may be different pitches. In 19-ET (or ~1/3-comma meantone
temperament), for example, the difference is as large as the change
in pitch made by adding a sharp or flat to a note.

What makes the augmented 2nd dissonant (as opposed to the minor 3rd)
is the fact that the tones are heard in (a harmonic) context as 1
scale step apart (instead of 2).

> > As Paul suggested, you should play around with a harmonic minor
scale
> > to hear this for yourself.
>
> I have tried, but I don't find anything on my 12-et piano...
> probably because I don't know how to create the musical context
> that would produce the effect that you both are describing.
> Where is Paul's high school music teacher when we need her? :)

Maybe the best way to illustrate this is with an example in the key
of C minor:

If you play:

C B----- C
G Ab G F Eb

I believe that you will hear a resolution from a dominant G to a
tonic C minor. Even though there are no full chords, the harmony is
implied by the context of the composition in which this example might
occur. Now let's add another voice above that to fill in the missing
notes so as to make the harmony explicit rather than implied:

Eb D----- G
C B----- C
G Ab G F Eb

Now suppose that you change the upper line to this:

C C----- C
G Ab G F Eb

Now you'll hear a subdominant F minor to a tonic C minor, in which
the Ab does not sound dissonant, even when you add notes to fill out
the implied harmony:

Eb F G Ab G
C C------ C
G Ab G F Eb

Now consider this third example, also in the key of C minor:

F Eb--- D
B C B C B

Now you should hear a (dissonant) diminished 5th resolving to a
(consonant) minor 3rd , followed by a (dissonant) diminished 4th.
This is made clear if another line is added to supply the notes that
are implied:

F Eb--- D
B C B C B
G G---- G

In this last example the (dissonant) diminished 4th is, in 12-ET, the
same size as a (consonant) major 3rd, but because of the implied
harmony, it is not heard as such, since the two tones are 3 diatonic
scale steps apart (not 2). So here (in example 3) we have a
consonant C:Eb moving to a dissonant B:Eb contrasted with (in example
1) a dissonant Ab:B moving to a consonang G:B -- in 12-ET the
intervals are exactly the same sizes, but the harmonic context makes
all the difference.

Does this help?

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

[snip]
> Maybe the best way to illustrate this is with an example in the key
> of C minor:
>
> If you play:
>
> C B----- C
> G Ab G F Eb
>
> I believe that you will hear a resolution from a dominant G to a
> tonic C minor. Even though there are no full chords, the harmony
is
> implied by the context of the composition in which this example
might
> occur. Now let's add another voice above that to fill in the
missing
> notes so as to make the harmony explicit rather than implied:
>
> Eb D----- G
> C B----- C
> G Ab G F Eb
>
> Now suppose that you change the upper line to this:
>
> C C----- C
> G Ab G F Eb
>
> Now you'll hear a subdominant F minor to a tonic C minor, in which
> the Ab does not sound dissonant, even when you add notes to fill
out
> the implied harmony:
>
> Eb F G Ab G
> C C------ C
> G Ab G F Eb
>
> Now consider this third example, also in the key of C minor:
>
> F Eb--- D
> B C B C B
>
> Now you should hear a (dissonant) diminished 5th resolving to a
> (consonant) minor 3rd , followed by a (dissonant) diminished 4th.
> This is made clear if another line is added to supply the notes
that
> are implied:
>
> F Eb--- D
> B C B C B
> G G---- G
>
> In this last example the (dissonant) diminished 4th is, in 12-ET,
the
> same size as a (consonant) major 3rd, but because of the implied
> harmony, it is not heard as such, since the two tones are 3
diatonic
> scale steps apart (not 2). So here (in example 3) we have a
> consonant C:Eb moving to a dissonant B:Eb contrasted with (in
example
> 1) a dissonant Ab:B moving to a consonang G:B -- in 12-ET the
> intervals are exactly the same sizes, but the harmonic context
makes
> all the difference.
>
> Does this help?
>
> --George

Yes it does. Thanks!
The same size intervals really do sound different depending on
the context, but I have to get my mind set to believe that I am
now playing something in C minor. I guess that is why it didn't
work when I tried before, because I was just playing intervals.

--Jeff