the multidimensional tuning

Hello!
While I'm new in this e-group, I still haven't read, what particular
kinds of ideas (obviously, concerning tuning) are mostly discussed
here, but I think, some people could be interested in the things I
write about here (or that somebody will point me useful links or other
information in the case I'm wrong). I, myself interest in the
possibility to expand tuning (meaning mostly for programmed
synthesizers or musical sound synthesis software, but not necessary)
to 2-dimensional, adding yet one parameter, in order to make sound
more natural and to solve at least the problem of naturalness of sound
(i. e. the "natural" sounding versus the "synthetic" one). (This
problem has one very strange aspect: A musician, listening a "natural"
recording through a phone, that cuts up all higher harmonics, still
says the music is natural, but a programmer can't achieve the natural
sounding via synthesis even having all harmonics, that present in the
best recordings. The controversy was tried to solve, saying that it
means, that naturalness originates in chaos. But I disagree, that
such regular sensation, that is naturalness of musical sound {not of
sound in general, which naturalness, e g. reverberations in a room,
has already been synthesized successfully}, could be caused by totally
irregular processes, especially that people identify this naturalness
with harmony and perfection[!]). And I think, that this problem lies
more in theoretic than in practical synthesis, and any new ideas
(including the throughly forgotten old) could be useful. What concerns
the idea of multidimensional tuning concretely, I've written an
abstract "The 2-dimensional tuning. A mathematical model", where I put
my ideas on it in mathematical form(You may read it from
http://www.geocities.com/linasrim/2-dtuning.pdf ). The abstract also
contains preface with more detailed explaining, what I speak about.
Now, if somebody interests in this idea (either in general, or in my
version that's in the abstract), I think, we may start a discussion or
simply to share information on it.
Yet , if anybody is interested, but can't read my abstract from the
address above, please write me, and I'll send You the file via e-mail,
whenever I have known about.

Linas

--- In tuning@yahoogroups.com, "Linas Plankis" <linasrim@y...> wrote:

> ... I've written an abstract "The 2-dimensional tuning.
> A mathematical model", where I put my ideas on it in
> mathematical form (You may read it from
> http://www.geocities.com/linasrim/2-dtuning.pdf ).

I'd be interested if the mathematicians here would read this
and comment on it, before i invest the time in trying to
understand it. Thanks.

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

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> --- In tuning@yahoogroups.com, "Linas Plankis" <linasrim@y...> wrote:
>
> > ... I've written an abstract "The 2-dimensional tuning.
> > A mathematical model", where I put my ideas on it in
> > mathematical form (You may read it from
> > http://www.geocities.com/linasrim/2-dtuning.pdf ).
>
>
>
> I'd be interested if the mathematicians here would read this
> and comment on it, before i invest the time in trying to
> understand it. Thanks.
>

OK. Hmm - not sure whether _I_ understand it right, though...
Linas Plankis introduces a second dimension (the first dimension is
just pitch, as we know it), which he calls "hardness" and which, to
me, appears to be more a matter of timbre. Sure, tuning and timbre are
related - yet I would hesitate to call this a tuning. I find a little
confusing that hardness is also called a "frequency" - I assume this
is to emphasize the "common concept" or something. Sometimes indeed
timbre is connected to frequencies, such as modulator frequencies in
FM synthesis or cutoff frequencies of a low-pass filter - but not
always, so OI would call this word not appropriate, if this model
really is to be a starting point for a more general concept, which
seems to be the claim Linas makes.

In the second part, several concrete "hardness" formulas are derived,
one of which measures the deviation of the pitches of a given scale
relative to an equal tempered scale. Now this again would have not
much to do with timbre - so this appears to be quite a different thing
than the timbre stuff in the beginning. Hence I must be
misunderstanding something - which may be my fault or Linas's...
--
Hans Straub

--- hstrhstraubhstrhstraubeletelesonique> wrote:

> --- In tuning@yahoyahoogroups, "monzmonzonzmonz.>
> wrote:
> >
> > --- In tuning@yahoyahoogroups, "LinaLinasnPlankis
<linalinasrim.> wrote:
> >
> > > ... I've written an abstract "The 2-dimensional
> tuning.
> > > A mathematical model", where I put my ideas on
> it in
> > > mathematical form (You may read it from
> > >
httphttpww.wwwcgeocities/linalinasrimtundtuning pdf).
> >
> >
> >
> > I'd be interested if the mathematicians here would
> read this
> > and comment on it, before i invest the time in
> trying to
> > understand it. Thanks.
> >
>
> OK. Hmm Hmmot sure whether _I_ understand it right,
> though...
> LinaLinasnPlankisroduces a second dimension (the
> first dimension is
> just pitch, as we know it), which he calls
> "hardness" and which, to
> me, appears to be more a matter of timbre. Sure,
> tuning and timbre are
> related - yet I would hesitate to call this a
> tuning. I find a little
> confusing that hardness is also called a "frequency"
> - I assume this
> is to emphasize the "common concept" or something.
> Sometimes indeed
> timbre is connected to frequencies, such as
> modulator frequencies in
> FM synthesis or cutoff frequencies of a low-pass
> filter - but not
> always, so OI would call this word not appropriate,
> if this model
> really is to be a starting point for a more general
> concept, which
> seems to be the claim LinaLinases.
>
> In the second part, several concrete "hardness"
> formulas are derived,
> one of which measures the deviation of the pitches
> of a given scale
> relative to an equal tempered scale. Now this again
> would have not
> much to do with timbre - so this appears to be quite
> a different thing
> than the timbre stuff in the beginning. Hence I
> must be
> misunderstanding something - which may be my fault
> or LinaLinas's
> --
> Hans StraStraub
It's OK. I see, You understood weak sides both of the
2-dimensional tuning in general and subsequently of my
conception. I also thought like you proposed here for
long time, but, later, I tried to cut this Gordian
node the simplest way. It seems me, that conception of
timbre and pure pitch has both acoustic and
psycpsychoacousticects. But does
psycpsychoacousticinitions of timbre and pitch
distinction are really identical to physically
acoustic ones? I know that they should be very very
similar, but i doubt if someone could ensure an ideal
identity here. And it's the real problem of quality
of, for example, midi performance, which raises this
question of the identity.
I think so, and I should defend this, but there's yet
few things, that allow to skip this problem at all,
not discussing over it. The 2-dimensional tuning
theory (in my version) is useful even in the case we
don't recognize objectivity of the 2nd dndension.
Someone may call the theory 1-dimensional with an
additional fictive parameter in this case, but i gave
the name 2-dimensional, for this name seems more
understandable and obvious. Saying, that it's useful,
i mean, that the theory provides new approach for
classifying (of gamuts, ethnic tunings etc.) that
could be helpful in some cases.
And yet. The second dimension is called hardness in
the theory. The term is conditional (at least by many
aspects) and it doesn't mean directly the timbre
hardness as one hears it. It's a process with
frequency, that's called this way, and not vice
versversat the timbre hardness is suggested to have
frequency here. I could call it differently than
hardness, but almost all words have connotations with
something known. So, hardness is simply a name for the
2nd dndension and it doesn't coincide with timbre
hardness.
Thanks for the ideas and for Your interest.

LinaLinasnPlankisP. S. What concerns measuring the
deviations of the pitches of a given scale relative to
an equal tempered scale, there's an important moment:
the idea of it is, that the results of measuring
should be assigned to the second dimension, if we want
to hear something constructive (�harmonic�). Or, in
other words, it's supposed here, that this measuring
of the deviations is more essential characteristic of
tuning than many others (i don't think that everybody
will agree with me at a moment, but, i think, raising
of the question, what priorities of characteristics of
tuning scales are, is reasoned).

__________________________________
Yahoo! Mail - PC Magazine Editors' Choice 2005
http://mail.yahoo.com

--- hstraub64 <hstraub64@telesonique.net> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "Linas Plankis"
> <linasrim@y...> wrote:
> >
> > > ... I've written an abstract "The 2-dimensional
> tuning.
> > > A mathematical model", where I put my ideas on
> it in
> > > mathematical form (You may read it from
> > > http://www.geocities.com/linasrim/2-dtuning.pdf
> ).
> >
> >
> >
> > I'd be interested if the mathematicians here would
> read this
> > and comment on it, before i invest the time in
> trying to
> > understand it. Thanks.
> >
>
> OK. Hmm - not sure whether _I_ understand it right,
> though...
> Linas Plankis introduces a second dimension (the
> first dimension is
> just pitch, as we know it), which he calls
> "hardness" and which, to
> me, appears to be more a matter of timbre. Sure,
> tuning and timbre are
> related - yet I would hesitate to call this a
> tuning. I find a little
> confusing that hardness is also called a "frequency"
> - I assume this
> is to emphasize the "common concept" or something.
> Sometimes indeed
> timbre is connected to frequencies, such as
> modulator frequencies in
> FM synthesis or cutoff frequencies of a low-pass
> filter - but not
> always, so OI would call this word not appropriate,
> if this model
> really is to be a starting point for a more general
> concept, which
> seems to be the claim Linas makes.
>
> In the second part, several concrete "hardness"
> formulas are derived,
> one of which measures the deviation of the pitches
> of a given scale
> relative to an equal tempered scale. Now this again
> would have not
> much to do with timbre - so this appears to be quite
> a different thing
> than the timbre stuff in the beginning. Hence I
> must be
> misunderstanding something - which may be my fault
> or Linas's...
> --
> Hans Straub
>
>
>
>

It's OK. I see, You understood weak sides both of the
2-dimensional tuning in general and subsequently of my
conception. I also thought like you proposed here for
long time, but, later, I tried to cut this Gordian
node the simplest way. It seems me, that conception of
timbre and pure pitch has both acoustic and
psychoacoustic aspects. But does psychoacoustic
definitions of timbre and pitch distinction are really
identical to physically acoustic ones? I know that
they should be very very similar, but i doubt if
someone could ensure an ideal identity here. And it's
the real problem of quality of, for example, midi
performance, which raises this question of the
identity.
I think so, and I should defend this, but there's yet
few things, that allow to skip this problem at all,
not discussing over it. The 2-dimensional tuning
theory (in my version) is useful even in the case we
don't recognize objectivity of the 2nd dimension.
Someone may call the theory 1-dimensional with an
additional fictive parameter in this case, but i gave
the name 2-dimensional, for this name seems more
understandable and obvious. Saying, that it's useful,
i mean, that the theory provides new approach for
classifying (of gamuts, ethnic tunings etc.) that
could be helpful in some cases.

And yet. The second dimension is called hardness in
the theory. The term is conditional (at least by many
aspects) and it doesn't mean directly the timbre
hardness as one hears it. It's a process with
frequency, that's called this way, and not vice versa,
not the timbre hardness is suggested to have frequency
here. I could call it differently than hardness, but
almost all words have connotations with something
known. So, hardness is simply a name for the 2nd
dimension and it doesn't coincide with timbre
hardness.
Thanks for the ideas and for Your interest.

Linas Plankis

P. S. What concerns measuring the deviations of the
pitches of a given scale relative to an equal tempered
scale, there's an important moment: the idea of it is,
that the results of measuring should be assigned to
the second dimension, if we want to hear something
constructive (�harmonic�). Or, in other words, it's
supposed here, that this measuring of the deviations
is more essential characteristic of tuning than many
others (i don't think that everybody will agree with
me at a moment, but, i think, raising of the question,
what priorities of characteristics of tuning scales
are, is reasoned).

__________________________________
Yahoo! Mail - PC Magazine Editors' Choice 2005
http://mail.yahoo.com

Hi Linas,

On Thu, 13 Oct 2005, you wrote:

... [much snipt]

> ... i don't think that everybody will agree with me at a moment ...

I am sorry to say this: I simply don't understand you.

So I cannot state whether I agree with you, or not.

But I want to understand your meaning more plainly. Do you
have any ideas on that?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
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--- Yahya Abdal-Aziz <yahya@melbpc.org.au> wrote:

>
> Hi Linas,
>
> On Thu, 13 Oct 2005, you wrote:
>
> ... [much snipt]
>
> > ... i don't think that everybody will agree with
> me at a moment ...
>
> I am sorry to say this: I simply don't understand
> you.
>
> So I cannot state whether I agree with you, or not.
>
> But I want to understand your meaning more plainly.
> Do you
> have any ideas on that?
>
> Regards,
> Yahya
>
> --

Hi, Yahya
Perhaps I had to write this:

PREHISTORY

I tried to apply different tunings for midi
performance, but i met some problems doing it:

1.Whatever you do, many people still say, that
sounding of midi recordings is �unnatural�. Perhaps we
shouldn't pay attention to what ones say, but I've
heard the same estimation from people who were far
from admiring classical music or disliking electronic
devices. And this situation raises necessary question,
why one couldn't synthesize sound, that was according
with preferences and tastes of the people, that think
the way, I described. And where's a problem, when both
synthetic sound and the sound of �natural� recordings
consists the same way of 44100 numbers per second? And
how it concerns possibilities of tuning?
2.When we make different tunings we need to classify
them some way. I found, that nobody can say
definitely, what parameters of a tuned scale (as
number of sounds in the octave, deviation from well
tempered scales and so on) are more essential for good
classifying
3.The similar problem is with a prognosis what a
tuning will be like, when we know parameters of a
scale, having not heard sound yet. Perhaps somebody
knows more about it, but I didn't find how to know
properties of a group of certain scales (when choosing
the one necessary) without long hours of testing.
4.And the so called ethnic tunings also are worth to
think about. Why do the same pitch divisions cause
such different musical effects (compare, for example,
the Western European heptatonics with the Southern
Asian one, or the Eastern Asian pentatonics with the
North American one /meaning the pentatonics that's
discussed up right now here/). How the same division
can have such different consequences for music
composition?

THE ABSTRACT

I didn't find good answers to these questions, and
decided to create myself, what helped me to solve them
at least partially. I don't know, if anybody else has
similar ideas, that's why I wrote to the tuning group.
Some of my ideas i put to the abstract �the
two-dimensional tuning�
(http://www.geocities.com/linasrim/2-dtuning.pdf )

Now, the main idea of what i wrote, was, that we
(including the earlier myself) make a mistake,
thinking, that process of tuning is a division of
one-dimensional range of pitch. When it actually
should be two-dimensional. This idea is very similar,
but i couldn't choose anything instead of this,
because all one-dimensional ideas have been already
tried. Many of them were tried also by people, who
participate in the group here.
And what background ideas do we have, to say, that
tuning should be two dimensional?:

1.The first background idea is , that our inner
imagination of pitch isn't one-dimensional. What do i
mean? We imagine a pitch, storing at last two
parameters for each sound (that is, for pure pitch
representing without yet other musical parameters) in
our memory. Where we may know it from? It isn't a
difficult thing. Tuning or singing, we divide
continued line of frequencies into tones. But any such
division , or discretization, needs yet one parameter
to do it.
2.The next idea is, that since earlier times, there
were cues about two-dimensional character of tuning.
The well known example of these cues is our musical
notation system. It obviously could be simpler for
one-dimensional representing (for example, having
simply 12 steps for each octave), but the additional
possibilities here are cues to two-dimensional
character of tuning (these additional possibilities
are skipped, for example, in the standard midi scale,
thus reducing it to one-dimensional indeed). An
example of these additional possibilities is the fact,
that C-sharp and D-flat, for example, are supposed
categorically different sounds in the system with no
or minimal pitch difference, and nobody doubted, that
such distinction is real, for long time.

Now, if i say, that tuning has two dimensions, i
should say, how to find these two dimensions. Some
things are said in my abstract
(http://www.geocities.com/linasrim/2-dtuning.pdf ),
but i see now, that they aren't said sufficiently
clearly. But the main idea is like this:
I take acoustic frequency as the first dimension. And
i take another parameter, let's say parameter x, as
the second dimension. Say, we don't know, what the
next dimension is like to. But we can try to find
something, that is obvious or that is similarity of
truth. Modeling this way, skipping less acceptable
possibilities and assuming more acceptable, it's
possible to achieve interesting results. The main of
these is the rule (it's called the second law of the
harmonic tuning in the abstract), that says, that for
harmonic tuning value of the second dimension equals
to deviation of the pitch frequencies ( that is, of
the first dimension values) of a given scale (relative
to an equal tempered scale) raised by a certain
degree.
Without other interesting possibilities this rule has
the consequence, that deviation of the pitch
frequencies of a given scale relative to an equally
tempered scale is the main characteristic of a given
scale (meaning, that tunings like 12-EDO are more sets
of necessary scales than musical scales themselves).
And it works. After making this conclusion we could
even skip the dimension x and use the pitch frequency
only, but already knowing the rule.
But if anybody wants to try possibilities of the
adding the x-parameter to real tuning, there is a
restriction. Such experiments aren't possible, using
using software that works exclusively with midi data,
nor with any EMI-s or other devices, that have
predefined one-dimensional tuning scale. Such
experiments are possible with the synthesizing
programs like csound or clm.

SO, i tried to answer Your question, but not knowing,
if i did. I afraid that there could be other things
that need explaining. And i'll be glad, if You give me
some hint, what isn't clear more concretely.

Thanks for Your interest.
Linas

__________________________________
Yahoo! Music Unlimited
Access over 1 million songs. Try it free.
http://music.yahoo.com/unlimited/

Hello, Linas,

> The second dimension is called hardness in the theory. The term is
conditional (at least by many aspects) and it doesn't mean directly
the timbre hardness as one hears it. It's a process with frequency,
that's called this way, and not vice versa, not the timbre hardness is
suggested to have frequency here. I could call it differently than
hardness, but almost all words have connotations with something known.
So, hardness is simply a name for the 2nd dimension and it doesn't
coincide with timbre hardness.

Montal's 1830s instructions about piano tuning rested upon words like
hard and soft besides sharp and flat (it makes a nice allusion to hard
and soft wires that sound like their names), but I think also there is
an established translation about old tetrachord divisions called hard
and soft.

Maybe that first is in some sense related, these kinds of second
dimensions in tuning. It is familiar that the standard tuning for
piano is in fact not exactly standard because of the behavior in each
instrument. Although this is from timbre it is not so easily heard
like in a bell, tuning sets so that the imagined temperament is
modified by some varying amounts to sound better (either to the tuner
or else to the musician, it is not decided!). In fact there is a
modern French piano tuning patenting such things so that the equal
temperament has sharper fifths (perhaps also the same is the Steinway
tuning, of course it will be different used on a Pleyel). If we are
lucky this dimension that can be calculated with some accuracy is used
constructing the instrument and afterwards tuning machines can
interpolate a pattern using few measurements, although one newer
device will measure notes dynamically because the sound changes with
the pitch being tuned, otherwise guided hearing organized series of
comparisons. But also the roughness from simultaneous notes tuned to
the same difference changes with their pitches, if there is some
timbre so that a tuner may like to adjust making intervals sound
better in different registers. There are also studies about sine waves
suggesting preference of sharpened octaves that sometimes might
explain why it is common finding highest notes in pianos tuned higher
than where they sound strongest, in a way that they exceed adjustment
from the mechanical inharmonicity (other times it might be from those
studies directly, built into some tuning machines and maybe some
schools).

Clark

Hi again Linas,

On Mon, 17 Oct 2005, you (Linas Plankis) wrote:
>
> --- Yahya Abdal-Aziz <yahya@...> wrote:
> >
> > Hi Linas,
> >
> > On Thu, 13 Oct 2005, you wrote:
> >
> > ... [much snipt]
> >
> > > ... i don't think that everybody will agree with
> > me at a moment ...
> >
> > I am sorry to say this: I simply don't understand
> > you.
> >
> > So I cannot state whether I agree with you, or not.
> >
> > But I want to understand your meaning more plainly.
> > Do you
> > have any ideas on that?
> >
> > Regards,
> > Yahya
> >
> > --
>
> Hi, Yahya
> Perhaps I had to write this:
>
> PREHISTORY
>
> I tried to apply different tunings for midi
> performance, but i met some problems doing it:
>
> 1.Whatever you do, many people still say, that
> sounding of midi recordings is �unnatural�. Perhaps we
> shouldn't pay attention to what ones say, but I've
> heard the same estimation from people who were far
> from admiring classical music or disliking electronic
> devices. And this situation raises necessary question,
> why one couldn't synthesize sound, that was according
> with preferences and tastes of the people, that think
> the way, I described. And where's a problem, when both
> synthetic sound and the sound of �natural� recordings
> consists the same way of 44100 numbers per second? And
> how it concerns possibilities of tuning?

So two reasons for your theory are: an attempt to
understand why some people regard MIDI recordings
as less satisfactory than other kinds of sound recording,
including electronic music - and to begin to remedy that.

> 2.When we make different tunings we need to classify
> them some way. I found, that nobody can say
> definitely, what parameters of a tuned scale (as
> number of sounds in the octave, deviation from well
> tempered scales and so on) are more essential for good
> classifying

A good classification, surely, is one that suits your needs
and your purposes. If this is true, then programs like
Manuel Op de Coul's Scala do go a long way towards
giving us the information, based on which we can classify
tunings and scales in very many different ways. The
problem, as I see it, is not a lack of information but
simply a lack of direction. We must ask ourselves:
What do we want this tuning to do for us?

> 3.The similar problem is with a prognosis what a
> tuning will be like, when we know parameters of a
> scale, having not heard sound yet. Perhaps somebody
> knows more about it, but I didn't find how to know
> properties of a group of certain scales (when choosing
> the one necessary) without long hours of testing.

You're impatient? Look at how many centuries, millenia
of accumulated musical experience is summarised in the
knowledge made available by using resources such as
Scala, or Joe Monzo's "Tonalsoft Encyclopedia of
Microtonal Music Theory". Surely it's worth spending
a few hours or days to review this material and see
what use you can make of it?

> 4.And the so called ethnic tunings also are worth to
> think about. Why do the same pitch divisions cause
> such different musical effects (compare, for example,
> the Western European heptatonics with the Southern
> Asian one, or the Eastern Asian pentatonics with the
> North American one /meaning the pentatonics that's
> discussed up right now here/). How the same division
> can have such different consequences for music
> composition?

I hope you're not overlooking the obvious - music is
arguably just as much about time as about pitch - some
say more. For you can have a distinctly musical
experience involving rhythmic drumming, without any
clear pitch or scale being involved. Contrariwise, few
of us feel that it is music to play several pitches drawn
from a scale randomly, with no regard to organising
those pitches in time.

The less obvious point to make here is that - as we have
recently seen in the discussion of blues tunings on this
list - although two different musical cultures may use
exactly the same scale as a formal basis, their micro-
tonal interpretation of that scale may be very different.
Indeed, some kinds of music, based largely on those
instruments which can readily be played at fixed pitch,
evolve styles in which purity of pitch and timbre become
goals, whilst other cultures pursue a contrary goal of
maximum expressiveness of intonation through varying
pitch and timbre. Contrast, for instance, the norms of
European opera traditions with those of Chinese opera
traditions; both of which rely heavily on the use of the
(heptatonic) diatonic scale. Or contrast the concert
flute with the shakuhachi. The point I'm making here
is that the "official" or theoretical scale is very often
_not_ the scale as played in actual practice by the best
musicians.

>
> THE ABSTRACT
>
> I didn't find good answers to these questions, and
> decided to create myself, what helped me to solve them
> at least partially. I don't know, if anybody else has
> similar ideas, that's why I wrote to the tuning group.
> Some of my ideas i put to the abstract �the
> two-dimensional tuning�
> (http://www.geocities.com/linasrim/2-dtuning.pdf )
>
> Now, the main idea of what i wrote, was, that we
> (including the earlier myself) make a mistake,
> thinking, that process of tuning is a division of
> one-dimensional range of pitch. When it actually
> should be two-dimensional.

What still strikes me as very strange at this point,
is that although you say there should be two dimensions
to tuning - yet you can't name or describe the second
one. To me this seems to be an idea you haven't yet got
very clear in your own mind. And why should there not
be seven dimensions instead?

> ... This idea is very similar,
> but i couldn't choose anything instead of this,
> because all one-dimensional ideas have been already
> tried. Many of them were tried also by people, who
> participate in the group here.
> And what background ideas do we have, to say, that
> tuning should be two dimensional?:
>
> 1.The first background idea is , that our inner
> imagination of pitch isn't one-dimensional. What do i
> mean? We imagine a pitch, storing at last two
> parameters for each sound (that is, for pure pitch
> representing without yet other musical parameters) in
> our memory. Where we may know it from? It isn't a
> difficult thing. Tuning or singing, we divide
> continued line of frequencies into tones. But any such
> division , or discretization, needs yet one parameter
> to do it.

So you're saying we can divide the continuum of
frequencies at any point, to choose the pitches of our
tuning? Yes, we do this. But this involves a single
dimension - more exactly, a specific measurable
quantity, namely the fundamental frequency of the
notes produced by our musical instruments. There is
no other dimension, no other measurable quantity,
which we use when specifying a tuning. If you say
there SHOULD be, then what is that dimension?
We measure frequency in Hertz; in what units can we
measure your new dimension?

> 2.The next idea is, that since earlier times, there
> were cues about two-dimensional character of tuning.
> The well known example of these cues is our musical
> notation system.

Which clearly represents, chiefly, pitch against time.
Pitch is the dimension of tuning, whilst time is the
dimension of meter and rhythm. Meter is not a tuning
phenomenon, nor is rhythm. So our musical notation
system represents pitch in one dimension only.

> ... It obviously could be simpler for
> one-dimensional representing (for example, having
> simply 12 steps for each octave), but the additional
> possibilities here are cues to two-dimensional
> character of tuning (these additional possibilities
> are skipped, for example, in the standard midi scale,
> thus reducing it to one-dimensional indeed). An
> example of these additional possibilities is the fact,
> that C-sharp and D-flat, for example, are supposed
> categorically different sounds in the system with no
> or minimal pitch difference, and nobody doubted, that
> such distinction is real, for long time.

Only in specially constructed tuning systems, such as
12-EDO, can we regard C# and Db as the same note
for some - though not all - purposes. In most historical
tuning systems, they are indeed set at different
pitches. In such systems, there is a difference of
value between the two notes on the single dimension of
tuning - namely pitch. You still haven't shown a need
for a second dimension.

>
> Now, if i say, that tuning has two dimensions, i
> should say, how to find these two dimensions. Some
> things are said in my abstract
> (http://www.geocities.com/linasrim/2-dtuning.pdf ),

I've tried repeatedly to look this up, but am having
difficulties with my internet connection. I will try
again if you think that would help me understand
your meaning better.

> ... but i see now, that they aren't said sufficiently
> clearly. But the main idea is like this:
> I take acoustic frequency as the first dimension. And
> i take another parameter, let's say parameter x, as
> the second dimension. Say, we don't know, what the
> next dimension is like to. But we can try to find
> something, that is obvious or that is similarity of
> truth. Modeling this way, skipping less acceptable
> possibilities and assuming more acceptable, it's
> possible to achieve interesting results. The main of
> these is the rule (it's called the second law of the
> harmonic tuning in the abstract), that says:

I would like to understand this "law":
> ... that for
> harmonic tuning value of the second dimension equals
> to deviation of the pitch frequencies ( that is, of
> the first dimension values) of a given scale (relative
> to an equal tempered scale) raised by a certain
> degree.
... but I find it doesn't actually make sense as a
sentence of the English language. Should there
perhaps be a comma after "harmonic tuning"? And may
I try paraphrasing a little? If so, your law might read -

"For harmonic tuning, the value of the second dimension
equals the deviation of the pitch frequencies (that is, of
the first dimension values) of a given scale (from an
equal tempered scale with the same number of notes)
raised to a fixed degree."

So, we might have a 12-EDO scale consisting of:
C 0 (cents) 0
C# 100 0
D 200 0
D# 300 0
E 400 0
F 500 0
F# 600 0
G 700 0
G# 800 0
A 900 0
A# 1000 0
B 1100 0
C' 1200 0

and a 12-UDO scale consisting of:
C 0 (cents) 0 (cents^2)
C# 110 100
D 200 0
D# 320 400
E 400 0
F 500 0
F# 630 900
G 700 0
G# 840 1600
A 900 0
A# 1050 2500
B 1100 0
C' 1200 0

if we chose for our fixed power the number 2.

How could we use the second column of figures
- the squared deviations from equal, in cents
squared - to differentiate this tuning from
another one? Say, one in which the second
dimension used the power 0.5?

Whatever value I give to that power, I'm at a
loss to understand how I could use that second
column. Particularly when you say it's "for
harmonic tuning". As iunderstand it, "harmonic
tuning" ought to mean "tuning so that some
notes are harmonics ie overtones of other
notes". But your deviation-power law doesn't
give harmonics or overtones, which are notes
whose frequencies are simple integer multiples.

> Without ...

[considering]?

> ... other interesting possibilities this rule has
> the consequence, that deviation of the pitch
> frequencies of a given scale relative to an equally
> tempered scale is the main characteristic of a given
> scale (meaning, that tunings like 12-EDO are more sets
> of necessary scales than musical scales themselves).
> And it works.

??? How?

> ... After making this conclusion we could
> even skip the dimension x and use the pitch frequency
> only, but already knowing the rule.

OK, suppose we skip the second dimension, and use only
the fundamental frequency of a note. (This is what the
rest of us still do.) What does "knowing the rule" mean?
That somehow the (squared, say) deviations of the
tuning from 12-EDO are useful harmonically? I think I
showed above that they are not.

> But if anybody wants to try possibilities of the
> adding the x-parameter to real tuning, there is a
> restriction. Such experiments aren't possible, using
> using software that works exclusively with midi data,
> nor with any EMI-s or other devices, that have
> predefined one-dimensional tuning scale. Such
> experiments are possible with the synthesizing
> programs like csound or clm.

Suppose I use a MIDI-capable compositional tool,
such as NoteWorthy Composer. With this, I create
MIDI files, which I then use to play my MIDI
keyboard. That keyboard incorporates a whole pile of
sampled instruments in its hardware. In the MIDI file,
I include pitch-bends as well as standard MIDI note
numbers, so that I can sweep through all frequencies
from the lowest MIDI note to the highest. Because
the keyboard has a high degree of polyphony, I can
layer many instruments simultaneously.

Now suppose I use a soft synth like csound, or even
a hardware synth such as my old Moog. I will grant
you that these synths can create some fairly simple
sounds that I can't get from the MIDI keyboard.
But almost any of the more complex sounds they
make can be approximated on the MIDI keyboard.
This is as true of their pitch as of their timbre.

So why can't I do the same experiments with either
setup? Probably more important - how do you
propose I should go about doing such experiments,
say in csound, when you can't tell me what parameter
I need to change? What are the units of measure of
this fictitious dimension?

>
> SO, i tried to answer Your question, but not knowing,
> if i did. I afraid that there could be other things
> that need explaining. And i'll be glad, if You give me
> some hint, what isn't clear more concretely.

Well, I've tried to show you why I still don't
understand you. I hope this helps you clarify your
concepts and shows you some points that need more
explanation.
>
> Thanks for Your interest.
> Linas
>
Thanks for the effort you took to reply in such detail.

Regards,
Yahya

--
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Hi Yahya,

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> What still strikes me as very strange at this point,
> is that although you say there should be two dimensions
> to tuning - yet you can't name or describe the second
> one. To me this seems to be an idea you haven't yet got
> very clear in your own mind. And why should there not
> be seven dimensions instead?

Is that a sly allusion to my work? ;-)

> > ... This idea is very similar,
> > but i couldn't choose anything instead of this,
> > because all one-dimensional ideas have been already
> > tried. Many of them were tried also by people, who
> > participate in the group here.
> > And what background ideas do we have, to say, that
> > tuning should be two dimensional?:
> >
> > 1.The first background idea is , that our inner
> > imagination of pitch isn't one-dimensional. What do i
> > mean? We imagine a pitch, storing at last two
> > parameters for each sound (that is, for pure pitch
> > representing without yet other musical parameters) in
> > our memory. Where we may know it from? It isn't a
> > difficult thing. Tuning or singing, we divide
> > continued line of frequencies into tones. But any such
> > division , or discretization, needs yet one parameter
> > to do it.
>
> So you're saying we can divide the continuum of
> frequencies at any point, to choose the pitches of our
> tuning? Yes, we do this. But this involves a single
> dimension - more exactly, a specific measurable
> quantity, namely the fundamental frequency of the
> notes produced by our musical instruments. There is
> no other dimension, no other measurable quantity,
> which we use when specifying a tuning. If you say
> there SHOULD be, then what is that dimension?
> We measure frequency in Hertz; in what units can we
> measure your new dimension?
>
>
>
> > 2.The next idea is, that since earlier times, there
> > were cues about two-dimensional character of tuning.
> > The well known example of these cues is our musical
> > notation system.
>
> Which clearly represents, chiefly, pitch against time.
> Pitch is the dimension of tuning, whilst time is the
> dimension of meter and rhythm. Meter is not a tuning
> phenomenon, nor is rhythm. So our musical notation
> system represents pitch in one dimension only.

It always strikes me that tuning theorists so often
seem to forget that pitch is *also* dependent upon time.
Remember, Hz is the same as cycles-per-*second*.

Thanks for asking all these questions of Linas, as i
am very interested in understanding his theory too,
but don't grok much of the math.

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

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> On Mon, 17 Oct 2005, you (Linas Plankis) wrote:
> >
> >
>
> > 2.The next idea is, that since earlier times, there
> > were cues about two-dimensional character of tuning.
> > The well known example of these cues is our musical
> > notation system.
>
> Which clearly represents, chiefly, pitch against time.
> Pitch is the dimension of tuning, whilst time is the
> dimension of meter and rhythm. Meter is not a tuning
> phenomenon, nor is rhythm. So our musical notation
> system represents pitch in one dimension only.
>

Not exactly! Observe that in classical music notation, there are 1)
the vertical positions of the notes and 2) accidentals before the
notes. Those can be seen as two dimensions, the first indicating the
position in units of a certain diatonic scale and the second the
deviation of this one. The word "sharpness" would not be inappropriate!

Physically, however, there is clearly only one dimension (pitch
height) - the second dimension is merely a matter of thinking/harmonic
context. Which is relevant, of course! Not for midi performance,
though - that's why I still think there are two quite different things
in Linas' paper not necessarily having much in common.

> > ... but i see now, that they aren't said sufficiently
> > clearly. But the main idea is like this:
> > I take acoustic frequency as the first dimension. And
> > i take another parameter, let's say parameter x, as
> > the second dimension. Say, we don't know, what the
> > next dimension is like to. But we can try to find
> > something, that is obvious or that is similarity of
> > truth. Modeling this way, skipping less acceptable
> > possibilities and assuming more acceptable, it's
> > possible to achieve interesting results. The main of
> > these is the rule (it's called the second law of the
> > harmonic tuning in the abstract), that says:

> I would like to understand this "law":

In my opinion, the term "second law of harmonic tuning" is a little
too "big" for it - I would not consider it so fundamental. But that is
just my opinion.

BTW, for those who have access to Mazzola's "The Topos of Music": the
process of adding dimensions is treated there, under the keyword as
"address change". One-dimensional tuning is referenced to as
"zero-addressed", and two-dimensional as "self-addressed" -
terminology is not to everyone's taste, I admit. I do not know the
details very well, but AFAIK Thomas Noll has done a bunch of
investigations around "self-addressed" harmony.
--
Hans Straub

--- threesixesinarow <CACCOLA@NET1PLUS.COM> wrote:

> Hello, Linas,
>
> > The second dimension is called hardness in the
> theory. The term is
> conditional (at least by many aspects) and it
> doesn't mean directly
> the timbre hardness as one hears it. It's a process
> with frequency,
> that's called this way, and not vice versa, not the
> timbre hardness is
> suggested to have frequency here. I could call it
> differently than
> hardness, but almost all words have connotations
> with something known.
> So, hardness is simply a name for the 2nd dimension
> and it doesn't
> coincide with timbre hardness.
>
> Montal's 1830s instructions about piano tuning
> rested upon words like
> hard and soft besides sharp and flat (it makes a
> nice allusion to hard
> and soft wires that sound like their names), but I
> think also there is
> an established translation about old tetrachord
> divisions called hard
> and soft.
>
> Maybe that first is in some sense related, these
> kinds of second
> dimensions in tuning. It is familiar that the
> standard tuning for
> piano is in fact not exactly standard because of the
> behavior in each
> instrument. Although this is from timbre it is not
> so easily heard
> like in a bell, tuning sets so that the imagined
> temperament is
> modified by some varying amounts to sound better
> (either to the tuner
> or else to the musician, it is not decided!). In
> fact there is a
> modern French piano tuning patenting such things so
> that the equal
> temperament has sharper fifths (perhaps also the
> same is the Steinway
> tuning, of course it will be different used on a
> Pleyel). If we are
> lucky this dimension that can be calculated with
> some accuracy is used
> constructing the instrument and afterwards tuning
> machines can
> interpolate a pattern using few measurements,
> although one newer
> device will measure notes dynamically because the
> sound changes with
> the pitch being tuned, otherwise guided hearing
> organized series of
> comparisons. But also the roughness from
> simultaneous notes tuned to
> the same difference changes with their pitches, if
> there is some
> timbre so that a tuner may like to adjust making
> intervals sound
> better in different registers. There are also
> studies about sine waves
> suggesting preference of sharpened octaves that
> sometimes might
> explain why it is common finding highest notes in
> pianos tuned higher
> than where they sound strongest, in a way that they
> exceed adjustment
> from the mechanical inharmonicity (other times it
> might be from those
> studies directly, built into some tuning machines
> and maybe some
> schools).
>
> Clark
>
>
>
>
Hello, Clark.

Thanks for the useful examples, You gave.
I, for example, myself, often think, that the human
hearing of sound, namely the part of it, that is
called �subjective� has some compensation mechanisms,
that collate dynamics, microreverberations, certain
elements of timbre and the fundamental frequency to
one phenomenon, that we call pitch (in the
psychoacoustic sense). But all this is in microlevel
only, so we hear greater changes of timbre, dynamics,
etc as timbre, dynamics, but smaller steps already
affect compensation mechanisms, being collated to one
pitch.
But the recognition, that pitch of hearing is composed
from different acoustic elements doesn't make a
mathematical theory. So, i see my theory as an example
or as an attempt, but this attempt seems not far from
the truth. But the person, who will completely
describe, what the dimension �x� completely means
acoustically, and if we should be satisfied with 2
dimensions only, that person will be a great
discoverer. Such person should be also an excellent
tuning master (meaning traditional tuning) and so on
and also be keen in electronic acoustics.
I only think, that the dimension �x� exists, and
tuning shouldn't have more than 2 dimensions, because
it would be very difficult to manage the structure in
the case. But the x dimension can without many doubts
be composed from few acoustic elements.

Linas

__________________________________
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Hi, Yahya.
Thanks for Your questions. Your counterarguments are
known for me. And i didn't call, what i said,
arguments or proofs, saying only they are questions to
think and cues. If I had a non controversial non
refutable proof, I would put it in the initial section
of all this.
For example, speaking about ethnic music, You say,
that
> hope you're not overlooking the obvious - music is
>arguably just as much about time as about pitch -
some
>say more. For you can have a distinctly musical
>experience involving rhythmic drumming, without any
>clear pitch or scale being involved. Contrariwise,
few
>of us feel that it is music to play several pitches
drawn
>from a scale randomly, with no regard to organising
>those pitches in time.
No, I actually am not overlooking this fact, and even
could add, that not every compositional aspect, that
involves pitch, is caused by character of tuning.
There are other things, including both compositional
theories and cultural background of a particular
nation. But some facts prompt to think about different
understanding of tuning, even having the same scale of
fundamental frequencies.
Now, i'd prefer to reply to each Your argument, but
one of them needs to be answered first. I assum, that
others are still hanging in this case.
You write:
>I would like to understand this "law":
>> ... that for
>> harmonic tuning value of the second dimension
equals
>> to deviation of the pitch frequencies ( that is, of
>> the first dimension values) of a given scale
(relative
>> to an equal tempered scale) raised by a certain
>> degree.
>... but I find it doesn't actually make sense as a
>sentence of the English language. Should there
>perhaps be a comma after "harmonic tuning"? And may
>I try paraphrasing a little? If so, your law might
read -

>"For harmonic tuning, the value of the second
dimension
>equals the deviation of the pitch frequencies (that
is, of
>the first dimension values) of a given scale (from an
>equal tempered scale with the same number of notes)
>raised to a fixed degree."

>So, we might have a 12-EDO scale consisting of:
>C 0 (cents) 0
>C# 100 0
>D 200 0
>D# 300 0
>E 400 0
>F 500 0
>F# 600 0
>G 700 0
>G# 800 0
>A 900 0
>A# 1000 0
>B 1100 0
>C' 1200 0
>
>and a 12-UDO scale consisting of:
>C 0 (cents) 0 (cents^2)
>C# 110 100
>D 200 0
>D# 320 400
>E 400 0
>F 500 0
>F# 630 900
>G 700 0
>G# 840 1600
>A 900 0
>A# 1050 2500
>B 1100 0
>C' 1200 0
>
>if we chose for our fixed power the number 2.
>
>How could we use the second column of figures
>- the squared deviations from equal, in cents
>squared - to differentiate this tuning from
>another one? Say, one in which the second
>dimension used the power 0.5?
>
>Whatever value I give to that power, I'm at a
>loss to understand how I could use that second
>column. Particularly when you say it's "for
>harmonic tuning". As iunderstand it, "harmonic
>tuning" ought to mean "tuning so that some
>notes are harmonics ie overtones of other
>notes". But your deviation-power law doesn't
>give harmonics or overtones, which are notes
>whose frequencies are simple integer multiples.

Yes, the definition, that i gave here for the formula,
is too weak in the sense of English (particularly �by
degree�). It couldn't lead to any other thing than
misunderstanding. We compare simply two scales here.
If we compare pure frequencies we divide each value
from a scale, that is under analysis, by its
respective value from the EDO with the same number of
sounds. If we compare logarithmic values, as values in
cents, we should subtract instead of dividing. Also
the scaler exists, which is an index of degree in the
case with pure frequencies and is simple multiplier in
the case with cents. But we can not use the scaler at
all, when we get results for analysis of a scale only.
It's necessary in the context of other parts of theory
only. So, the formula will be
X(n) = (P(n) � Pet(n))*a.
(X(n) is the dimension X, the hardness as I've call it
in the abstract, P(n) is the n-th value of fundamental
frequency of the searched scale and Pet(n) is the
value of fundamental frequency of the EDO with the
same number of sounds, while a is the scaler)
We may skip 'a', as i've said, and change 'Pet(n)' to
'1200*n/m' (where m is a number of sounds in the
searched scale), getting:
X(n) = P(n) � 1200*n/m
Now, having the formula, which representing should be
more clear than my English, we can do the same what
You tried. But there's one thing. I think, we should
distinguish compositional scales and general scales
here (I do this distinction in my theory). For it's
not serious to talk about harmonic tuning of a general
scale, which actually is a set of scales to choose.
Harmonic or not harmonic can only be a scale, that is
used for composing. So, i'll take the heptatonics and
C-dur as a particular case of it, and the C-dur will
be based on 12-EDO. If You prefer 12-tonics, we may
calculate a respective example later.
Now, The first column contains relative frequency
values in cents:
C 0 (cents)
D 200
E 400
F 500
G 700
A 900
B 1100
C 1200

And values of frequency of the 7-EDO are these:
0(C) 0 (cents)
1(D) 171.4
2(E) 342.9
3(F) 514.3
4(G) 685.7
5(A) 857.1
6(B) 1028.6
7(C) 1200

The difference will be:
C 0 (cents)
D 28.6
E 57.1
F -14.3
G 14.3
A 42.9
B 71.4
C 0

And this column gives good background data for
analysis of the scale. For example, you may draw a
graph (with number in the x and result in the y).
Now, imagine, that we look at shape of the graph.
Number of peaks, growing or descending of X(n) and
other aspects of the distribution are significant,
analyzing the scale. And i'm almost ensured from my
experiments, that, if we analyze a scale this way,
similar shapes give always similar sounding and
similar compositional effects. Obviously one may
analyze any other scale this way. I myself yet group
scales into pattern scales and not pattern, or
secondary, scales. The pattern scales are those, that
have similar range of X(n) values to the range of the
C-dur (actually i use always the same range, and
different value of the scaler, but it's the same). The
C-dur is chosen, because it's perhaps the best tested
scale in the history of mankind, but it's conditional.

And yet. Perhaps the proposed procedure itself isn't
new, but i tried to give not only the procedure , but
a different approach too.
The other answers to be continued, as i've wrote.
Linas.

__________________________________
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--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

> BTW, for those who have access to Mazzola's "The Topos of Music": the
> process of adding dimensions is treated there, under the keyword as
> "address change". One-dimensional tuning is referenced to as
> "zero-addressed", and two-dimensional as "self-addressed" -
> terminology is not to everyone's taste, I admit.

Using topoi in this connection is using a thermonuclear device to swat
a fly.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> It always strikes me that tuning theorists so often
> seem to forget that pitch is *also* dependent upon time.

Yes, and hence they forget that describing the pitches and amplitudes
of a set of notes vary through time does *not* provide a unique
representation of the corresponding sound. The classical uncertainty
principle, which is a purely mathematical result, tends to get
neglected as a result of this forgetting.

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
> >
> >
> > On Mon, 17 Oct 2005, you (Linas Plankis) wrote:
> > >
> > >
> >
> > > 2.The next idea is, that since earlier times, there
> > > were cues about two-dimensional character of tuning.
> > > The well known example of these cues is our musical
> > > notation system.
> >
> > Which clearly represents, chiefly, pitch against time.
> > Pitch is the dimension of tuning, whilst time is the
> > dimension of meter and rhythm. Meter is not a tuning
> > phenomenon, nor is rhythm. So our musical notation
> > system represents pitch in one dimension only.
> >
>
> Not exactly! Observe that in classical music notation, there are 1)
> the vertical positions of the notes and 2) accidentals before the
> notes. Those can be seen as two dimensions, the first indicating
the
> position in units of a certain diatonic scale and the second the
> deviation of this one.

Exactly! And this is how we hear Western music, according to some
academics -- even if G# and Ab are tuned the same way, they fall in
completely different places in this two-dimensional representation,
and thus are *heard* as different pitches. Context, it seems,
determines which one is heard in a particular place.

On Wed, 19 Oct 2005, "hstraub64" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> >
> > On Mon, 17 Oct 2005, you (Linas Plankis) wrote:
> > >
> > >
> >
> > > 2.The next idea is, that since earlier times, there
> > > were cues about two-dimensional character of tuning.
> > > The well known example of these cues is our musical
> > > notation system.
> >
> > Which clearly represents, chiefly, pitch against time.
> > Pitch is the dimension of tuning, whilst time is the
> > dimension of meter and rhythm. Meter is not a tuning
> > phenomenon, nor is rhythm. So our musical notation
> > system represents pitch in one dimension only.
> >
>
> Not exactly! Observe that in classical music notation, there are 1)
> the vertical positions of the notes and 2) accidentals before the
> notes. Those can be seen as two dimensions, the first indicating the
> position in units of a certain diatonic scale and the second the
> deviation of this one. The word "sharpness" would not be inappropriate!
>
> Physically, however, there is clearly only one dimension (pitch
> height) - the second dimension is merely a matter of thinking/harmonic
> context. Which is relevant, of course! Not for midi performance,
> though - that's why I still think there are two quite different things
> in Linas' paper not necessarily having much in common.
>
> > > ... but i see now, that they aren't said sufficiently
> > > clearly. But the main idea is like this:
> > > I take acoustic frequency as the first dimension. And
> > > i take another parameter, let's say parameter x, as
> > > the second dimension. Say, we don't know, what the
> > > next dimension is like to. But we can try to find
> > > something, that is obvious or that is similarity of
> > > truth. Modeling this way, skipping less acceptable
> > > possibilities and assuming more acceptable, it's
> > > possible to achieve interesting results. The main of
> > > these is the rule (it's called the second law of the
> > > harmonic tuning in the abstract), that says:
>
> > I would like to understand this "law":
>
> In my opinion, the term "second law of harmonic tuning" is a little
> too "big" for it - I would not consider it so fundamental. But that is
> just my opinion.
>
> BTW, for those who have access to Mazzola's "The Topos of Music": the
> process of adding dimensions is treated there, under the keyword as
> "address change". One-dimensional tuning is referenced to as
> "zero-addressed", and two-dimensional as "self-addressed" -
> terminology is not to everyone's taste, I admit. I do not know the
> details very well, but AFAIK Thomas Noll has done a bunch of
> investigations around "self-addressed" harmony.
> --
> Hans Straub

Hi Hans,

"Sharpness", as represented by accidentals, is just
a notational device for adjusting the pitch of the note,
that is for most instruments, the frequency of the
fundamental component of the note. In this way it
does not differ at all from the device of placing notes
at different heights on a staff. We can readily see
this must be so, because, when playing an instrument
of fixed intonation, such as a piano or organ, if that
instrument is tuned to 12-EDO, a player must play
exactly the same note (strike exactly the same key)
when the notation says Cx as when it says D or Ebb.

So "sharpness" is just another name for "pitch" - the
single vertical dimension that staff notation sets off
against time in the single horizontal dimension.

Not having access to the Mazzola, I am unable to
refer to it for a further explanation of his meanings.
Certainly you have not managed to convey to me just
what his "extra dimensions" might be! :-)

Regards,
Yahya

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Hi all,

On Thu, 20 Oct 2005, "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, "hstraub64"
> <hstraub64@t...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:
> > >
> > >
> > > On Mon, 17 Oct 2005, you (Linas Plankis) wrote:
> > > >
> > > >
> > >
> > > > 2.The next idea is, that since earlier times, there
> > > > were cues about two-dimensional character of tuning.
> > > > The well known example of these cues is our musical
> > > > notation system.
> > >
> > > Which clearly represents, chiefly, pitch against time.
> > > Pitch is the dimension of tuning, whilst time is the
> > > dimension of meter and rhythm. Meter is not a tuning
> > > phenomenon, nor is rhythm. So our musical notation
> > > system represents pitch in one dimension only.
> > >
> >
> > Not exactly! Observe that in classical music notation, there are 1)
> > the vertical positions of the notes and 2) accidentals before the
> > notes. Those can be seen as two dimensions, the first indicating
> > the
> > position in units of a certain diatonic scale and the second the
> > deviation of this one.
>
> Exactly! And this is how we hear Western music, according to some
> academics -- even if G# and Ab are tuned the same way, they fall in
> completely different places in this two-dimensional representation,
> and thus are *heard* as different pitches. Context, it seems,
> determines which one is heard in a particular place.

"Some academics"?! Come, Paul, you can do better than that!
Why not give us all a link, or a book title (with author and date,
of course) :-)

None of what you or wrote above, or Hans wrote earlier,
convinces me of any possible need for a "second dimension"
of tuning.

How can the deviation of a pitch from some notional ideal,
measured in pitch units, occupy anything but a pitch dimension?

And what would you make of a listener (from Erewhon?) who
didn't possess this neo-Platonic ideal of a diatonic (or chromatic,
or 53-EDO, or pentatonic, etc) scale?

To be fair, I'll try to suspend disbelief for a moment. Then -

If this IS a real phenomenon - that we hear deviations from
an ideal scale in a different way from hearin notes of that
scale, then of course it applies to the blues as well - perhaps
our "pentatonic hearing" is the basis against which we hear
"blueness"? So "blueness" might be a good name, or perhaps
"chromaticity", for Linas Plankis' Dimension X.

Actually, I think I have a better name for Dimension X.
I've managed to download and read Linas' PDF article. His
example, given there, assigns each note of the Pythagorean
heptatonic scale an X value that simply counts the number
of fifths it is removed from the note C, then normalises that
number by dividing it by 7. So Dimension X seems to be just
another name for (one measure of) Harmonic Distance.

Regards,
Yahya

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Hi Linas,

On Wed, 19 Oct 2005, Linas Plankis wrote:
> Hi, Yahya.
> Thanks for Your questions. Your counterarguments are
> known for me. And i didn't call, what i said,
> arguments or proofs, saying only they are questions to
> think and cues. If I had a non controversial non
> refutable proof, I would put it in the initial section
> of all this.

I understand, you are rather suggesting another way
of thinking about tuning than the customary single
dimension of pitch. You merely choose to use the
terminology and methods of maths since they reflect
the nature of your thinking, if not their exact form.

> For example, speaking about ethnic music,

A slight correction - I was speaking of all music,
rather than just ethnic music.

> ...You say,
> that
> > hope you're not overlooking the obvious - music is
> >arguably just as much about time as about pitch -
> some
> >say more. For you can have a distinctly musical
> >experience involving rhythmic drumming, without any
> >clear pitch or scale being involved. Contrariwise,
> few
> >of us feel that it is music to play several pitches
> drawn
> >from a scale randomly, with no regard to organising
> >those pitches in time.
> No, I actually am not overlooking this fact, and even
> could add, that not every compositional aspect, that
> involves pitch, is caused by character of tuning.
> There are other things, including both compositional
> theories and cultural background of a particular
> nation. But some facts prompt to think about different
> understanding of tuning, even having the same scale of
> fundamental frequencies.

So what is _objectively_ the same tuning may have
entirely different _subjective_ uses, depending on one's
compositional theories and cultural background? Yes,
I couldn't agree more! A system of tuning is but a
skeleton to support an entire musical body, whose sinews
may be rhythm, muscles may be gesture, face may be
expression and clothes may be decoration ...

> Now, i'd prefer to reply to each Your argument, but
> one of them needs to be answered first. I assum, that
> others are still hanging in this case.

Makes sense to start at the most basic question.

> You write:
> >I would like to understand this "law":
> >> ... that for
> >> harmonic tuning value of the second dimension
> equals
> >> to deviation of the pitch frequencies ( that is, of
> >> the first dimension values) of a given scale
> (relative
> >> to an equal tempered scale) raised by a certain
> >> degree.
> >... but I find it doesn't actually make sense as a
> >sentence of the English language. Should there
> >perhaps be a comma after "harmonic tuning"? And may
> >I try paraphrasing a little? If so, your law might
> read -
>
> >"For harmonic tuning, the value of the second
> dimension
> >equals the deviation of the pitch frequencies (that
> is, of
> >the first dimension values) of a given scale (from an
> >equal tempered scale with the same number of notes)
> >raised to a fixed degree."
>
> >So, we might have a 12-EDO scale consisting of:
> >C 0 (cents) 0
> >C# 100 0
> >D 200 0
> >D# 300 0
> >E 400 0
> >F 500 0
> >F# 600 0
> >G 700 0
> >G# 800 0
> >A 900 0
> >A# 1000 0
> >B 1100 0
> >C' 1200 0
> >
> >and a 12-UDO scale consisting of:
> >C 0 (cents) 0 (cents^2)
> >C# 110 100
> >D 200 0
> >D# 320 400
> >E 400 0
> >F 500 0
> >F# 630 900
> >G 700 0
> >G# 840 1600
> >A 900 0
> >A# 1050 2500
> >B 1100 0
> >C' 1200 0
> >
> >if we chose for our fixed power the number 2.
> >
> >How could we use the second column of figures
> >- the squared deviations from equal, in cents
> >squared - to differentiate this tuning from
> >another one? Say, one in which the second
> >dimension used the power 0.5?
> >
> >Whatever value I give to that power, I'm at a
> >loss to understand how I could use that second
> >column. Particularly when you say it's "for
> >harmonic tuning". As iunderstand it, "harmonic
> >tuning" ought to mean "tuning so that some
> >notes are harmonics ie overtones of other
> >notes". But your deviation-power law doesn't
> >give harmonics or overtones, which are notes
> >whose frequencies are simple integer multiples.
>
> Yes, the definition, that i gave here for the formula,
> is too weak in the sense of English (particularly �by
> degree�). It couldn't lead to any other thing than
> misunderstanding. We compare simply two scales here.
> If we compare pure frequencies we divide each value
> from a scale, that is under analysis, by its
> respective value from the EDO with the same number of
> sounds. If we compare logarithmic values, as values in
> cents, we should subtract instead of dividing. Also
> the scaler

A suggestion: perhaps you could use the expression
"scale multiplier" here in place of your word "scaler",
which people would be apt to confuse with a "scalar",
that is, non-vector or single-dimensional vector.

> exists, which is an index of degree in the
> case with pure frequencies and is simple multiplier in
> the case with cents. But we can not use the scaler at
> all, when we get results for analysis of a scale only.
> It's necessary in the context of other parts of theory
> only. So, the formula will be
> X(n) = (P(n) � Pet(n))*a.
> (X(n) is the dimension X, the hardness as I've call it
> in the abstract, P(n) is the n-th value of fundamental
> frequency of the searched scale and Pet(n) is the
> value of fundamental frequency of the EDO with the
> same number of sounds, while a is the scaler)
> We may skip 'a', as i've said, and change 'Pet(n)' to
> '1200*n/m' (where m is a number of sounds in the
> searched scale), getting:
> X(n) = P(n) � 1200*n/m
> Now, having the formula, which representing should be
> more clear than my English, we can do the same what
> You tried.

Good idea ...

> ... But there's one thing. I think, we should
> distinguish compositional scales and general scales
> here (I do this distinction in my theory). For it's
> not serious to talk about harmonic tuning of a general
> scale, which actually is a set of scales to choose.

OK - I guess you're making a distinction between:
1. An abstract scale, which is a set of interval ratios,
eg 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1
and
2. An actual or particular scale, which is a set of pitches,
given either as note names eg A B C# D E F# G# A, or
more precisely as fundamental frequencies eg
220 247.5 275 293.33 330 371.25 412.5 440.

Is this what you mean?

> Harmonic or not harmonic can only be a scale, that is
> used for composing. So, i'll take the heptatonics and
> C-dur as a particular case of it, and the C-dur will ...

Although the terms "dur" and "moll(e)" are mostly used in
European languages, we use "major" and "minor" almost
exclusively in English. So, let's say we're tallking about
the C major scale here.

> ... be based on 12-EDO. If You prefer 12-tonics, we may
> calculate a respective example later.
> Now, The first column contains relative frequency
> values in cents:
> C 0 (cents)
> D 200
> E 400
> F 500
> G 700
> A 900
> B 1100
> C 1200
>
> And values of frequency of the 7-EDO are these:
> 0(C) 0 (cents)
> 1(D) 171.4
> 2(E) 342.9
> 3(F) 514.3
> 4(G) 685.7
> 5(A) 857.1
> 6(B) 1028.6
> 7(C) 1200
>
> The difference will be:
> C 0 (cents)
> D 28.6
> E 57.1
> F -14.3
> G 14.3
> A 42.9
> B 71.4
> C 0
>
> And this column gives good background data for
> analysis of the scale. For example, you may draw a
> graph (with number in the x and result in the y).
> Now, imagine, that we look at shape of the graph.
> Number of peaks, growing or descending of X(n) and
> other aspects of the distribution are significant,
> analyzing the scale. And i'm almost ensured from my
> experiments, that, if we analyze a scale this way,
> similar shapes give always similar sounding and
> similar compositional effects.

I can believe that two heptatonic scales with _very_
similar shapes in their deviations-from-7from12EDO
curves would have both "similar sounds" and "similar
compositional effects". But as the similarity weakens,
I would expect the sounds to change, often quite
dramatically, and certainly in nonlinear ways. In fact,
I'd expect there to be a fairly small "tolerance" for
deviations in shape, perhaps dependent on the
sensitivity of the listener (as for Harmonic Entropy).
So I do think you're overstating the case here for
similarity of results.

> ... Obviously one may
> analyze any other scale this way. I myself yet group
> scales into pattern scales and not pattern, or
> secondary, scales. The pattern scales are those, that
> have similar range of X(n) values to the range of the
> C-dur (actually i use always the same range, and
> different value of the scaler, but it's the same). The
> C-dur is chosen, because it's perhaps the best tested
> scale in the history of mankind, but it's conditional.
>
> And yet. Perhaps the proposed procedure itself isn't
> new, but i tried to give not only the procedure , but
> a different approach too.
> The other answers to be continued, as i've wrote.
> Linas.

I look forward to your other answers!

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > Exactly! And this is how we hear Western music, according to some
> > academics -- even if G# and Ab are tuned the same way, they fall in
> > completely different places in this two-dimensional representation,
> > and thus are *heard* as different pitches. Context, it seems,
> > determines which one is heard in a particular place.
>
> "Some academics"?! Come, Paul, you can do better than that!
> Why not give us all a link, or a book title (with author and date,
> of course) :-)

Eytan Agmon, for one, though as Paul knows I think he fails to
understand the meaning and implications of his own theory. Agmon
suggests we understand Western music from a 12 and 7 point of view,
which to me means in effect from a 12&7 point of view. Introducing
that, a G# and an Ab are not the same. What these academics have
discovered, I think, is that the tradition of Western music is rooted
in this kind of thinking. Quelle surprise!

> None of what you or wrote above, or Hans wrote earlier,
> convinces me of any possible need for a "second dimension"
> of tuning.

It certainly makes sense to count the number of generators in a
regular tuning system, which introduces a dimensional factor. I think
Agmon's observation boils down to the claim that the rank-one group of
12-edo is treated as if it were a rank-two group. Being a good little
academic theorist, he refuses to see any tuning implications in such a
rank two group, despite the hundreds of years of history of Western
music which used rank two tuning, first Pythagorean and then meantone.
But of course thinking in those terms in this century is considered
heretical, and minds can close with an almost audible clanging noise.

Hi Yahya, Linas, Hans, and Paul,

I haven't really digested Linas's paper yet, and
have only skimmed briefly thru the discussion of it
here as it seems others have yet to understand it fully.

But from the beginning, this "two-dimensional" thing
has reminded me of Eric Regener's work, as presented in

Regener, Eric. 1973.
_Pitch notation and Equal Temperament: A Formal Study_

Almost 5 years ago to the day (my, how time flies),
i posted here the complete review of Regener's book
by Richard Chrisman:

/tuning/topicId_14954.html#14995

It's very long, and the formatting of the tables and
diagrams will be messed up on the Yahoo web interface.
I suggest hitting the "Reply" button, which will format
everything properly, then selecting the whole text of
my post and copying it into Notepad or some other text
editor. You might want to print it out for easier reading,
as it is long.

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

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > None of what you or wrote above, or Hans wrote earlier,
> > convinces me of any possible need for a "second dimension"
> > of tuning.
>
> It certainly makes sense to count the number of generators
> in a regular tuning system, which introduces a dimensional
> factor. I think Agmon's observation boils down to the claim
> that the rank-one group of 12-edo is treated as if it were
> a rank-two group. Being a good little academic theorist,
> he refuses to see any tuning implications in such a
> rank two group, despite the hundreds of years of history
> of Western music which used rank two tuning, first
> Pythagorean and then meantone.

I just want to be clear on how pythagorean can be rank two.
Is it this idea of 12&7 (i.e., simultaneously chromatic and
diatonic scales)?

It would be interesting to me to model pythagorean in
Tonescape on a 2-D lattice. Can you send me one?

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

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> I just want to be clear on how pythagorean can be rank two.
> Is it this idea of 12&7 (i.e., simultaneously chromatic and
> diatonic scales)?

The simplest way to see it is that Pythagorean is the same as 3-limit;
that is, everything is |a b>.

> It would be interesting to me to model pythagorean in
> Tonescape on a 2-D lattice. Can you send me one?

Tonescape represents octave classes; I'd need to figure out how to get
it to not do that.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > I just want to be clear on how pythagorean can be rank two.
> > Is it this idea of 12&7 (i.e., simultaneously chromatic and
> > diatonic scales)?
>
> The simplest way to see it is that Pythagorean is the same
> as 3-limit; that is, everything is |a b>.

Oh, right ... i'm so used to thinking of pythagorean as a
linear system that i forgot about the Identity Interval.
So since it uses factors 2 and 3, it's "rank two". Got it.

> > It would be interesting to me to model pythagorean in
> > Tonescape on a 2-D lattice. Can you send me one?
>
> Tonescape represents octave classes; I'd need to figure
> out how to get it to not do that.

Octave classes are required right now for Tonepaces
and for the ET wizard -- but not for Tunings.

You can create a new tuning using "File | New | Tuning"
and then specify that you don't want a Identity Interval,
or specify an Identity Interval of any size you'd like,
using a cents-value.

There's an example of a Tonescape Tuning without an
Identity Interval, which you can upload from our website
-- Archytas's Enharmonic:

http://tonalsoft.com/downloads/free-tunings.aspx

And there's a small Tonescape Score illustrating each
of the 5 tetrachords of the Perfect Immutable System (PIS)
as tuned in Archytas's Enharmonic:

http://tonalsoft.com/downloads/free-scores.aspx

The tutorial on creating a Tuning is here:

http://tonalsoft.com/support/tonescape/tut-tuning.aspx

Call me on the phone if you need more help.

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

> "For harmonic tuning, the value of the second dimension
> equals the deviation of the pitch frequencies (that is, of
> the first dimension values) of a given scale (from an
> equal tempered scale with the same number of notes)
> raised to a fixed degree."
>
> So, we might have a 12-EDO scale consisting of:
> C 0 (cents) 0
> C# 100 0
> D 200 0
> D# 300 0
> E 400 0
> F 500 0
> F# 600 0
> G 700 0
> G# 800 0
> A 900 0
> A# 1000 0
> B 1100 0
> C' 1200 0
>
> and a 12-UDO scale consisting of:
> C 0 (cents) 0 (cents^2)
> C# 110 100
> D 200 0
> D# 320 400
> E 400 0
> F 500 0
> F# 630 900
> G 700 0
> G# 840 1600
> A 900 0
> A# 1050 2500
> B 1100 0
> C' 1200 0
>
> if we chose for our fixed power the number 2.
>
> How could we use the second column of figures
> - the squared deviations from equal, in cents
> squared - to differentiate this tuning from
> another one? Say, one in which the second
> dimension used the power 0.5?
>
> Whatever value I give to that power, I'm at a
> loss to understand how I could use that second
> column. Particularly when you say it's "for
> harmonic tuning". As iunderstand it, "harmonic
> tuning" ought to mean "tuning so that some
> notes are harmonics ie overtones of other
> notes". But your deviation-power law doesn't
> give harmonics or overtones, which are notes
> whose frequencies are simple integer multiples.

One intrepretation might be to play chords in
harmonic tuning, but let their roots be influenced
by the squared deviation from 12-tET. John
deLaubenfels did something similar with his
adaptune project (he later measured squared
deviation from COFT, his Calculated Optimal
Fixed Tuning, rather than from 12-tET). Hermode
tuning uses the sum of the unsquared deviations
from 12-tET in this way.

-Carl

> And this column gives good background data for
> analysis of the scale. For example, you may draw a
> graph (with number in the x and result in the y).
> Now, imagine, that we look at shape of the graph.
> Number of peaks, growing or descending of X(n) and
> other aspects of the distribution are significant,
> analyzing the scale. And i'm almost ensured from my
> experiments, that, if we analyze a scale this way,
> similar shapes give always similar sounding and
> similar compositional effects. Obviously one may
> analyze any other scale this way. I myself yet group
> scales into pattern scales and not pattern, or
> secondary, scales. The pattern scales are those, that
> have similar range of X(n) values to the range of the
> C-dur (actually i use always the same range, and
> different value of the scaler, but it's the same). The
> C-dur is chosen, because it's perhaps the best tested
> scale in the history of mankind, but it's conditional.

Hello Linas,

I can't follow the details of this paragraph... are
you saying that we could learn something about the
diatonic scale by looking at the deviations of its
tones from 7-tone equal temperament?

I wouldn't disagree, but I'm unsure how this could be
used to make more realistic MIDI performances.

I agree that MIDI music could benefit from more data
than standard MIDI files contain. But the MIDI
standard seems fairly extensible, if one were to write
a custom synthesis system.

Do you have any musical examples (audio or scores)?

-Carl

Hi Gene,

On Fri, 21 Oct 2005, "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > > Exactly! And this is how we hear Western music, according to some
> > > academics -- even if G# and Ab are tuned the same way, they fall in
> > > completely different places in this two-dimensional representation,
> > > and thus are *heard* as different pitches. Context, it seems,
> > > determines which one is heard in a particular place.
> >
> > "Some academics"?! Come, Paul, you can do better than that!
> > Why not give us all a link, or a book title (with author and date,
> > of course) :-)
>
> Eytan Agmon, for one, though as Paul knows I think he fails to
> understand the meaning and implications of his own theory. Agmon
> suggests we understand Western music from a 12 and 7 point of view,
> which to me means in effect from a 12&7 point of view.

Thanks, Gene.

Please explain what you mean by "a 12 and 7 point of view", and
"a 12&7 point of view". You say that the first means, to you, in
effect, the second. Is there any good reason to think they might
NOT be identical?

> ... Introducing
> that, a G# and an Ab are not the same. What these academics have
> discovered, I think, is that the tradition of Western music is rooted
> in this kind of thinking. Quelle surprise!
>
> > None of what you or wrote above, or Hans wrote earlier,
> > convinces me of any possible need for a "second dimension"
> > of tuning.
>
> It certainly makes sense to count the number of generators in a
> regular tuning system, which introduces a dimensional factor.

- one new dimension for each extra generator, including
the octave 2/1, right?

> I think
> Agmon's observation boils down to the claim that the rank-one group of
> 12-edo is treated as if it were a rank-two group. Being a good little
> academic theorist, he refuses to see any tuning implications in such a
> rank two group, despite the hundreds of years of history of Western
> music which used rank two tuning, first Pythagorean and then meantone.
> But of course thinking in those terms in this century is considered
> heretical, and minds can close with an almost audible clanging noise.

Not on this list, surely! :-)

Regards,
Yahya

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On Fri, 21 Oct 2005 "monz" wrote:

> Hi Yahya, Linas, Hans, and Paul,
>
> I haven't really digested Linas's paper yet, and
> have only skimmed briefly thru the discussion of it
> here as it seems others have yet to understand it fully.
>
> But from the beginning, this "two-dimensional" thing
> has reminded me of Eric Regener's work, as presented in
>
> Regener, Eric. 1973.
> _Pitch notation and Equal Temperament: A Formal Study_
>
> Almost 5 years ago to the day (my, how time flies),
> i posted here the complete review of Regener's book
> by Richard Chrisman:
>
> /tuning/topicId_14954.html#14995
>
> It's very long, and the formatting of the tables and
> diagrams will be messed up on the Yahoo web interface.
> I suggest hitting the "Reply" button, which will format
> everything properly, then selecting the whole text of
> my post and copying it into Notepad or some other text
> editor. You might want to print it out for easier reading,
> as it is long.

Monz,
I've saved your message as a pdf file, to study at leisure
(what leisure?!)

And in your reply:
/tuning/topicId_14954.html#15054
to Joe Pehrson's:
/tuning/topicId_14954.html#15006 ,
you asked for some help with the maths in the review.
Did the cavalry ever arrive?

Regards,
Yahya

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Hi Yahya,

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> On Fri, 21 Oct 2005 "monz" wrote:
>
> > Regener, Eric. 1973.
> > _Pitch notation and Equal Temperament: A Formal Study_
> >
> > Almost 5 years ago to the day (my, how time flies),
> > i posted here the complete review of Regener's book
> > by Richard Chrisman:
> >
> > /tuning/topicId_14954.html#14995
> >
>
>
> Monz,
> I've saved your message as a pdf file, to study at leisure
> (what leisure?!)
>
> And in your reply:
> /tuning/topicId_14954.html#15054
> to Joe Pehrson's:
> /tuning/topicId_14954.html#15006 ,
> you asked for some help with the maths in the review.
> Did the cavalry ever arrive?

Yes ... Paul Erlich to the rescue!

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

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > you asked for some help with the maths in the review.
> > Did the cavalry ever arrive?
>
>
> Yes ... Paul Erlich to the rescue!

In fact, it bore its fruit here:

http://tonalsoft.com/enc/t/transformation.aspx

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

Hi.

I put answers not exactly in the order, the questions
appeared. Some questions are more important than
others. I also try to answer the questions firstly,
that hint my accidental mistakes.

>> I didn't find good answers to these questions, and
>> decided to create myself, what helped me to solve
them
>> at least partially. I don't know, if anybody else
has
>> similar ideas, that's why I wrote to the tuning
group.
>> Some of my ideas i put to the abstract �the
>> two-dimensional tuning�
>> (http://www.geocities.com/linasrim/2-dtuning.pdf )
>>
>> Now, the main idea of what i wrote, was, that we
>> (including the earlier myself) make a mistake,
>> thinking, that process of tuning is a division of
>> one-dimensional range of pitch. When it actually
>> should be two-dimensional.
>
>What still strikes me as very strange at this point,
>is that although you say there should be two
dimensions
>to tuning - yet you can't name or describe the second
>one. To me this seems to be an idea you haven't yet
got
>very clear in your own mind. And why should there
not
>be seven dimensions instead?

We have two a bit different things here. When I say,
that our inner imagination of pitch is
two-dimensional, it could be a matter for one
discussion, but what concerns the theory, i offer
here, it could be discussed in another discussion.
It's possible to prove with high probability level,
that the first is true. But what concerns the theory,
it's connected not only with imaginative things, and
it perhaps raises more questions. Actually the idea,
that pitch is imagined in two dimensional way,
inspired to write the abstract. And the abstract has
ideas how it's possible, how it can be understood.
But even if the abstract contains something wrong, it
doesn't mean, that this inspirational idea is wrong.
The two-dimensionality of perceptional pitch is a
wider idea, than the theory itself.
Theoretically it's based on the understanding, that we
divide pitch and timbre very strictly in calculations,
basing on the Fourier analysis. But who can ensure,
that there's a Fourier analysis device in our brains.
Yes, the formula is very similar to what we heard, but
nobody can say that it's exactly the same. But it
would mean, that some aspects, that formally are part
of timbre, are heard as pitch. Now, let's take the
fundamental frequency as a number one and the
hypothetic rest of pitch, that is over the fundamental
frequency, as a number two. It cues two-dimensions
too.
If you've read my abstract more intently, you should
notice, that i simply used this. I divided the rest
from the fundamental frequency.
Now, why the number of dimensions is 2. An adding one
dimension may appear helpful, but if we added more, it
would cause too complicated system , as I think.
But i feel another question in this, for what purpose
the theory can be useful. I think:
1.It could serve as a pattern for those, who would
like experiment with how micro-timber elements of
sound affects pitch perception.
2.It could be used for classifying as we've discussed.

3.A theory could have no rational reason the same way
as any piece of art :-) (but it's a joke).

>> 2.The next idea is, that since earlier times, there
>> were cues about two-dimensional character of
tuning.
>> The well known example of these cues is our musical
>> notation system.
>
>Which clearly represents, chiefly, pitch against
time.
>Pitch is the dimension of tuning, whilst time is the
>dimension of meter and rhythm. Meter is not a tuning
>phenomenon, nor is rhythm. So our musical notation
>system represents pitch in one dimension only.
There are different points of view, why the notation
system has additional marks instead of having simply
12 positions for an octave. Ones may stress aspects of
historical traditions or even of economy. Looking
formally the system is two-dimensional, although the
second dimension has 3 positions only. But i don't
take it as an argument, it's a cue only.
Short answers
>> ... But there's one thing. I think, we should
>> distinguish compositional scales and general scales
>> here (I do this distinction in my theory). For it's
>> not serious to talk about harmonic tuning of a
general
>> scale, which actually is a set of scales to choose.

>OK - I guess you're making a distinction between:
>1. An abstract scale, which is a set of interval
ratios,
>eg 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1
>and
>2. An actual or particular scale, which is a set of
pitches,
>given either as note names eg A B C# D E F# G# A, or
>more precisely as fundamental frequencies eg
>220 247.5 275 293.33 330 371.25 412.5 440.
>
>Is this what you mean?
No, I meant a simpler thing. There are scales to tune
an instrument and there are scales, that are necessary
to perform a composition. Scales of the first kind
very often provide possibility to choose between few
scales of the second kind. The most known example is
12-EDO, when we use it not in the duodekatonics: it
can be useful to play C major, D major etc etc. I
simply want to say that just the second kind are
those, what I speak about here. That's because a scale
of the first kind is a bunch of scales indeed and it
naturally can't be analyzed the way I wrote (it hasn't
a single way of sounding!).

>Although the terms "dur" and "moll(e)" are mostly
used in
>European languages, we use "major" and "minor" almost
>exclusively in English. So, let's say we're tallking
about
>the C major scale here.

Thanks for the note. I'll revise, where it has a
sense.

A suggestion: perhaps you could use the _expression
"scale multiplier" here in place of your word
"scaler",
which people would be apt to confuse with a "scalar",
that is, non-vector or single-dimensional vector.

I agree. But perhaps �scaling multipyer�, shouldn't
it?

Linas

__________________________________
Yahoo! Mail - PC Magazine Editors' Choice 2005
http://mail.yahoo.com

>>> "For harmonic tuning, the value of the second
dimension
>>> equals the deviation of the pitch frequencies
(that is, of
>>> the first dimension values) of a given scale (from
an
>>> equal tempered scale with the same number of
notes)
>>> raised to a fixed degree."
>>
>> So, we might have a 12-EDO scale consisting of:
>> C 0 (cents) 0
>> C# 100 0
>> D 200 0
>> D# 300 0
>> E 400 0
>> F 500 0
>> F# 600 0
>> G 700 0
>> G# 800 0
>> A 900 0
>> A# 1000 0
>> B 1100 0
>> C' 1200 0
>>
>> and a 12-UDO scale consisting of:
>> C 0 (cents) 0 (cents^2)
>> C# 110 100
>> D 200 0
>> D# 320 400
>> E 400 0
>> F 500 0
>> F# 630 900
>> G 700 0
>> G# 840 1600
>> A 900 0
>> A# 1050 2500
>> B 1100 0
>> C' 1200 0
>>
Hello, Carl

It's a pity, but you took a wrong example written by
Yahya, not my own. My own is in the next to the
message that contains this one. So, no one may find
from this example anything about C major in the sense
that i mean.

But your question has a grain of truth. The first
thing, that I don't suppose a comparing of two
different sounds here. This formula gives a purely
mathematical operation that is based not on a real
equally tuned scale, but on mathematical form of it.
(It even contradicted to the theory, if we compared
two sounds. According to the theory every sound must
have yet dimension x, but i didn't said a word, what
dimension x of this equally tuned scale form is). So
one shouldn't understand the formula of harmonic
tuning like comparison, that raises a question of the
horizontal tuning. The grain of your truth is, that
the word 'harmony' perhaps isn't the best term in this
case. It should be understood in the general sense of
the word, as fitting, sympathy etc., and it simply
means fitting of two dimensions between themselves
here.

Linas

__________________________________
Yahoo! Mail - PC Magazine Editors' Choice 2005
http://mail.yahoo.com

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Please explain what you mean by "a 12 and 7 point of view", and
> "a 12&7 point of view". You say that the first means, to you, in
> effect, the second. Is there any good reason to think they might
> NOT be identical?

By "12 and 7" I mean a rank two tone group with the octave mapped to
(12, 7). By 12&7 I mean a system defined by the corresponding vals;
<12 19| and <7 11| for the 3-limit, and <12 19 28| and <7 11 16| for
the 5-limit. These have tuning implications; the first is just
3-limit, or Pythagorean, tuning. The second is meantone. Given that we
are talking about common practice music, it seems clear to me that we
are pretty well forced to assume that 12 and 7 means 12&7, ie
meantone, because the triad is an essential feature of common
practice, which initially based itself on meantone, and then modified
things a bit. But Agmon insists he wants to talk about this stuff
divorced from any tuning considerations; that is, to understand common
practice music in terms of a rank two tone group, but not to consider
heretical, old-fashioned tunings of that tone group to be relevant,
despite the history and the acoustical facts.

> > It certainly makes sense to count the number of generators in a
> > regular tuning system, which introduces a dimensional factor.
>
> - one new dimension for each extra generator, including
> the octave 2/1, right?

Correct.

> > But of course thinking in those terms in this century is considered
> > heretical, and minds can close with an almost audible clanging noise.
>
> Not on this list, surely! :-)

Academic theorists tend to blanch and run if the supremecy of 12edo is
questioned.

Hi Monz,

On Sun, 23 Oct 2005 "monz" wrote:
> Hi Yahya,
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> > On Fri, 21 Oct 2005 "monz" wrote:
> > > Regener, Eric. 1973.
> > > _Pitch notation and Equal Temperament: A Formal Study_
> > >
> > > Almost 5 years ago to the day (my, how time flies),
> > > i posted here the complete review of Regener's book
> > > by Richard Chrisman:
> > >
> > > /tuning/topicId_14954.html#14995
> > >
> >
> > Monz,
> > I've saved your message as a pdf file, to study at leisure
> > (what leisure?!)
> >
> > And in your reply:
> > /tuning/topicId_14954.html#15054
> > to Joe Pehrson's:
> > /tuning/topicId_14954.html#15006 ,
> > you asked for some help with the maths in the review.
> > Did the cavalry ever arrive?
>
> Yes ... Paul Erlich to the rescue!

...

> In fact, it bore its fruit here:
>
> http://tonalsoft.com/enc/t/transformation.aspx

Saved for later study.

Regards,
Yahya

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Hi Linas,

On Sun, 23 Oct 2005 Linas Plankis wrote:
> Hi.
>
> I put answers not exactly in the order, the questions
> appeared. Some questions are more important than
> others. I also try to answer the questions firstly,
> that hint my accidental mistakes.
>
> >> I didn't find good answers to these questions, and
> >> decided to create myself, what helped me to solve
> > > them
> >> at least partially. I don't know, if anybody else
> > > has
> >> similar ideas, that's why I wrote to the tuning
> > > group.
> >> Some of my ideas i put to the abstract �the
> >> two-dimensional tuning�
> >> (http://www.geocities.com/linasrim/2-dtuning.pdf )
> >>
> >> Now, the main idea of what i wrote, was, that we
> >> (including the earlier myself) make a mistake,
> >> thinking, that process of tuning is a division of
> >> one-dimensional range of pitch. When it actually
> >> should be two-dimensional.
> >
> >What still strikes me as very strange at this point,
> >is that although you say there should be two
> > dimensions
> >to tuning - yet you can't name or describe the second
> >one. To me this seems to be an idea you haven't yet
> > got
> >very clear in your own mind. And why should there
> > not
> >be seven dimensions instead?
>
> We have two a bit different things here. When I say,
> that our inner imagination of pitch is
> two-dimensional, it could be a matter for one
> discussion, but what concerns the theory, i offer
> here, it could be discussed in another discussion.
> It's possible to prove with high probability level,
> that the first is true.

I'm not really interested in what it's "possible to prove".
Just call me a Doubting Thomas, and SHOW me! :-)

> But what concerns the theory,
> it's connected not only with imaginative things, and
> it perhaps raises more questions. Actually the idea,
> that pitch is imagined in two dimensional way,
> inspired to write the abstract. And the abstract has
> ideas how it's possible, how it can be understood.
> But even if the abstract contains something wrong, it
> doesn't mean, that this inspirational idea is wrong.
> The two-dimensionality of perceptional pitch is a
> wider idea, than the theory itself.

Indeed, inspirations may be metaphorical rather than
literal.

> Theoretically it's based on the understanding, that we
> divide pitch and timbre very strictly in calculations,
> basing on the Fourier analysis. But who can ensure,
> that there's a Fourier analysis device in our brains.
> Yes, the formula is very similar to what we heard, but
> nobody can say that it's exactly the same. But it
> would mean, that some aspects, that formally are part
> of timbre, are heard as pitch. Now, let's take the
> fundamental frequency as a number one and the
> hypothetic rest of pitch, that is over the fundamental
> frequency, as a number two. It cues two-dimensions
> too.

So you're saying you DO intend us to interpret the
second dimension as timbre? I find this hard to take
literally. Since a timbre is a whole collection of
overtone frequencies at different amplitudes, it simply
cannot be represented by one dimension. There is no
natural ordering of timbres; the nearest we can come
to that is to create an arbitrary set of numbers to
index those timbres - along the lines of the General
MIDI patch numbers. (Tho I can see a dedicated
serialist might use exactly those numbers to create
a "timbre row", just as he uses scale degrees to create
a tone row.)

> If you've read my abstract more intently,

Very intently indeed. I though I understood that -
at least in your extended example in the abstract -
you identifed the second dimension as the harmonic
distance (measured in steps along the circle of fifths)
between two scale degrees.

> ... you should
> notice, that i simply used this. I divided the rest
> from the fundamental frequency.
> Now, why the number of dimensions is 2. An adding one
> dimension may appear helpful, but if we added more, it
> would cause too complicated system , as I think.

Not if that represented the nature of musical
reality better. Would our description of space-time
be better if we used only, say, time and radial distance?
Although that's perfectly adequate for some purposes
(eg "as the crow flies"), it is almost useless when we
need to describe the motions of, say, helicopters.
Essential complexity is not a thing to be avoided. If
an adequate description of phenomena requires 17
dimensions, then that's exactly how many we should
use. Otherwise, we will simply get the wrong results,
or no results at all.

> But i feel another question in this, for what purpose
> the theory can be useful. I think:
> 1.It could serve as a pattern for those, who would
> like experiment with how micro-timber elements of
> sound affects pitch perception.

Or with dynamics, or articulation, or spatial location
and direction of the instrument, or ... many other things
besides timbre; all affecting the kind of sound you
produce, in broad or in subtle ways.

> 2.It could be used for classifying as we've discussed.

I'm not clear what you mean here.

> 3.A theory could have no rational reason the same way
> as any piece of art :-) (but it's a joke).

Is it?

I think I've made clear what I think: the best purpose
for any theory of a system is to explain the realities
of that system: how it works, how it arises, how it
evolves, where its limits lie.

> >> 2.The next idea is, that since earlier times, there
> >> were cues about two-dimensional character of
> > tuning.
> >> The well known example of these cues is our musical
> >> notation system.
> >
> >Which clearly represents, chiefly, pitch against
> > time.
> >Pitch is the dimension of tuning, whilst time is the
> >dimension of meter and rhythm. Meter is not a tuning
> >phenomenon, nor is rhythm. So our musical notation
> >system represents pitch in one dimension only.

> There are different points of view, why the notation
> system has additional marks instead of having simply
> 12 positions for an octave. Ones may stress aspects of
> historical traditions or even of economy. Looking
> formally the system is two-dimensional, although the
> second dimension has 3 positions only. But i don't
> take it as an argument, it's a cue only.

One of the other list members implied recently that
we experience inflection - that is, performance
deviations from the basic scale, eg a JI diatonic scale -
in a way that differs from our perceptions of the scale
itself. I think there may be something to this. Whereas
a pianist or organist can only inflect the scale degrees
by semitones, a guitarist, fiddler or vocalist can bend or
smear them by microtones, and those bends can follow
any number of different curves. In certain modal styles
of music, those inflections become part of the musical
vocabulary, the permitted ornaments, and a means of
great expressive power. To analyse these ornaments
merely in terms of the nearest scale degrees is to miss
a great deal.

> Short answers
> >> ... But there's one thing. I think, we should
> >> distinguish compositional scales and general scales
> >> here (I do this distinction in my theory). For it's
> >> not serious to talk about harmonic tuning of a
> > > general
> >> scale, which actually is a set of scales to choose.
>
> >OK - I guess you're making a distinction between:
> >1. An abstract scale, which is a set of interval
> > ratios,
> >eg 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1
> >and
> >2. An actual or particular scale, which is a set of
> > pitches,
> >given either as note names eg A B C# D E F# G# A, or
> >more precisely as fundamental frequencies eg
> >220 247.5 275 293.33 330 371.25 412.5 440.
> >
> >Is this what you mean?
> No, I meant a simpler thing. There are scales to tune
> an instrument and there are scales, that are necessary
> to perform a composition. Scales of the first kind
> very often provide possibility to choose between few
> scales of the second kind. The most known example is
> 12-EDO, when we use it not in the duodekatonics: it
> can be useful to play C major, D major etc etc. I
> simply want to say that just the second kind are
> those, what I speak about here. That's because a scale
> of the first kind is a bunch of scales indeed and it
> naturally can't be analyzed the way I wrote (it hasn't
> a single way of sounding!).

Indeed. When you tune the piano's 88 notes, you
create a _gamut_ from which you can draw 2^88
different musical _scales_ for purposes of composition,
not all equally useful.

> >Although the terms "dur" and "moll(e)" are mostly
> > used in
> >European languages, we use "major" and "minor" almost
> >exclusively in English. So, let's say we're tallking
> > about
> >the C major scale here.
>
> Thanks for the note. I'll revise, where it has a
> sense.

Thanks. That will be clearer.

> A suggestion: perhaps you could use the _expression
> "scale multiplier" here in place of your word
> "scaler",
> which people would be apt to confuse with a "scalar",
> that is, non-vector or single-dimensional vector.
>
> I agree. But perhaps �scaling multipyer�, shouldn't
> it?

No.

Regards,
Yahya

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Gene,

On Sun, 23 Oct 2005, "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > Please explain what you mean by "a 12 and 7 point of view", and
> > "a 12&7 point of view". You say that the first means, to you, in
> > effect, the second. Is there any good reason to think they might
> > NOT be identical?
>
> By "12 and 7" I mean a rank two tone group with the octave mapped to
> (12, 7). By 12&7 I mean a system defined by the corresponding vals;
> <12 19| and <7 11| for the 3-limit, and <12 19 28| and <7 11 16| for
> the 5-limit.

So there's no unique interpretation of the name "m&n" for
arbitrary m and n; 12&7 is just a convenient shorthand for
all the numbers in those vals.

> These have tuning implications; the first is just
> 3-limit, or Pythagorean, tuning. The second is meantone. Given that we
> are talking about common practice music, it seems clear to me that we
> are pretty well forced to assume that 12 and 7 means 12&7, ie
> meantone, because the triad is an essential feature of common
> practice, which initially based itself on meantone, and then modified
> things a bit. But Agmon insists he wants to talk about this stuff
> divorced from any tuning considerations; that is, to understand common
> practice music in terms of a rank two tone group, but not to consider
> heretical, old-fashioned tunings of that tone group to be relevant,
> despite the history and the acoustical facts.

Thanks.

> > > It certainly makes sense to count the number of generators in a
> > > regular tuning system, which introduces a dimensional factor.
> >
> > - one new dimension for each extra generator, including
> > the octave 2/1, right?
>
> Correct.
>
> > > But of course thinking in those terms in this century is considered
> > > heretical, and minds can close with an almost audible clanging noise.
> >
> > Not on this list, surely! :-)
>
> Academic theorists tend to blanch and run if the supremecy of 12edo is
> questioned.

I've noticed a couple of academic email addresses here from
time to time ... Obviously some academics are more open-minded.

Regards,
Yahya

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> > Agmon suggests we understand Western music from a 12 and 7
> > point of view, which to me means in effect from a 12&7
> > point of view.
>
> Is there any good reason to think they might
> NOT be identical?

This is a good example of Gene assuming that readers of Tuning
would know an esoteric term or notation used on tuning math,
for which there is probably not even a definition anywhere that
could be looked up.

-Carl

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> I'm not really interested in what it's "possible to prove".
> Just call me a Doubting Thomas, and SHOW me! :-)

What I feel is lacking so far are precise definitions. There isn't
much point in postulating a second dimension if you cannot give a
clear statement of what it is.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > (12, 7). By 12&7 I mean a system defined by the corresponding vals;
> > <12 19| and <7 11| for the 3-limit, and <12 19 28| and <7 11 16| for
> > the 5-limit.
>
> So there's no unique interpretation of the name "m&n" for
> arbitrary m and n; 12&7 is just a convenient shorthand for
> all the numbers in those vals.

I use m&n to mean something specific (the rank two temperament defined
by two "standard" vals) so long as the prime limit is specified. Then
there's m+n, but that's a whole other story.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Hi Hans,
>
> "Sharpness", as represented by accidentals, is just
> a notational device for adjusting the pitch of the note,
> that is for most instruments, the frequency of the
> fundamental component of the note. In this way it
> does not differ at all from the device of placing notes
> at different heights on a staff. We can readily see
> this must be so, because, when playing an instrument
> of fixed intonation, such as a piano or organ, if that
> instrument is tuned to 12-EDO, a player must play
> exactly the same note (strike exactly the same key)
> when the notation says Cx as when it says D or Ebb.

And yet some composers do carefully observe these distinctions in
notation. Some theorists argue that these distinctions are actually
*heard*, because of context, and say it's naive to think that we hear
each note on the instrument as the same "grammatical" or "syntactic"
entity in the music even when it's spelled differently -- just as we
hear an augmented second as a very different interval from a minor
third, even though *physically*, absent context and musical meaning,
they are the same on a piano or organ.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> Some theorists argue that these distinctions are actually
> *heard*, because of context, and say it's naive to think that we hear
> each note on the instrument as the same "grammatical" or "syntactic"
> entity in the music even when it's spelled differently -- just as we
> hear an augmented second as a very different interval from a minor
> third, even though *physically*, absent context and musical meaning,
> they are the same on a piano or organ.

OK, I need to know: do *you* believe this is so, that someone will
actually hear an interval differently depending on context, even
though the measurement would be identical?

Cheers,
Jon

Hi, Yahya,
i continue answering to your questions:
>> ... other interesting possibilities this rule has
>> the consequence, that deviation of the pitch
>> frequencies of a given scale relative to an equally
>> tempered scale is the main characteristic of a
given
>> scale (meaning, that tunings like 12-EDO are more
sets
>> of necessary scales than musical scales
themselves).
>> And it works.
>
>??? How?
>
>
>> ... After making this conclusion we could
>> even skip the dimension x and use the pitch
frequency
>> only, but already knowing the rule.
>
>OK, suppose we skip the second dimension, and use
only
>the fundamental frequency of a note. (This is what
the
>rest of us still do.) What does "knowing the rule"
mean?
>That somehow the (squared, say) deviations of the
>tuning from 12-EDO are useful harmonically? I think
I
>showed above that they are not.

Perhaps you should draw a graph of deviations for
better visuality, but no one may insist on it. You
would see three moments, that are very obvious:
1.Graph has a number of peaks.
2.Steps of growing are different in size from steps of
descending.
3.And, without any doubt, the graph will have a
certain number of points.
These characteristics are essential for a scale. And
some secondary characteristics, as eg number of points
/ number of peaks, also could be successfully used.
Classifying is easy then. For example, we can take
scales, that have 17 points (are 17-tonics) and 3
peaks, as a group.
Also there's an interesting feature of scales. Some of
them have growing steps smaller than descending steps,
but others, in contrary, have growing steps bigger
than descending ones (compare diatonic and enharmonic
scales by Boetius / Aristoxenes). This feature is also
useful classifying scales.
How it works? Perhaps You should try: Create two
scales with similar or, better, same, characteristics
of the list above, except one. Perhaps it would be
better argument than all my arguing here.

The short answers:

>Indeed. When you tune the piano's 88 notes, you
>create a _gamut_ from which you can draw 2^88
>different musical _scales_ for purposes of
composition,
>not all equally useful.
Looking at the common practice period, one can see,
that the scale with 12 keys was used as a scale for
the heptatonics. This understanding is possible,
concerning not only the common practice period. I
simply exclude this taking of a super-scales for what
we speak about. And it's quite clear thing, so i take
this your idea more as a joke.

>> 2.It could be used for classifying as we've
discussed.
>I'm not clear what you mean here.
If the answer above isn't sufficient, the answer
still stands in the queue.

>> ... you should
>> notice, that i simply used this. I divided the rest
>> from the fundamental frequency.
>> Now, why the number of dimensions is 2. An adding
one
>> dimension may appear helpful, but if we added more,
it
>> would cause too complicated system , as I think.
>
>Not if that represented the nature of musical
>reality better. Would our description of space-time
>be better if we used only, say, time and radial
distance?
>Although that's perfectly adequate for some purposes
>(eg "as the crow flies"), it is almost useless when
we
>need to describe the motions of, say, helicopters.
>Essential complexity is not a thing to be avoided.
If
>an adequate description of phenomena requires 17
>dimensions, then that's exactly how many we should
>use. Otherwise, we will simply get the wrong
results,
>or no results at all.
If we discussed how to synthesize sound in generally,
i should agree with You. The same is about
synthesizing timbres for electronic instruments. But
when we speak about pitch, there are some limits. We
have our notational system at most two-dimensional,
and not 3- or more-dimensional, and it commits us to
be careful, especially when starting.

>Very intently indeed. I though I understood that -
>at least in your extended example in the abstract -
>you identifed the second dimension as the harmonic
>distance (measured in steps along the circle of
fifths)
>between two scale degrees.
The second dimension, as I understand it, isn't a
purely mathematical thing. If it was a result from the
formula only, what are we discussing about? I simply
would have to say, that �There's a formula, that helps
much in tuning. Let's try it�. But how the formula
works, anybody may try explain differently from me
too. E. g. like you say:
>One of the other list members implied recently that
>we experience inflection - that is, performance
>deviations from the basic scale, eg a JI diatonic
scale -
>in a way that differs from our perceptions of the
scale
>itself. I think there may be something to this.
However i don't think, that men hear deviations from
an (ET in our case) scale. I suppose, that the second
dimension should be proportional (taking logarithmic
values eg in cents) to the deviation, when we want to
hear specifically musical sounds.

Linas

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--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > Some theorists argue that these distinctions are actually
> > *heard*, because of context, and say it's naive to think that we
hear
> > each note on the instrument as the same "grammatical"
or "syntactic"
> > entity in the music even when it's spelled differently -- just as
we
> > hear an augmented second as a very different interval from a
minor
> > third, even though *physically*, absent context and musical
meaning,
> > they are the same on a piano or organ.
>
> OK, I need to know: do *you* believe this is so, that someone will
> actually hear an interval differently depending on context, even
> though the measurement would be identical?

Yes, absolutely. Several music teachers and professors have shocked
me with various demonstrations of this phenomenon (always in a
Western common-practice/tonal context). To the students in the class,
the augmented second sounds very different than a minor third; even
though they may physically be the same interval, the context that
fixes the "augmented second" or "minor third" interpretation has such
a strong influence on the way we hear that the two intervals don't
sound the same at all. For one thing, one sounds dissonant, while the
other sounds consonant. Without the ear-training that I've acquired
and undertaken, I'd have no idea that they were actually the same
physical interval.

The same thing is true, and much more widely acklowledged, in
language. What we perceive as the phonetic units of language are not
physically distinct sounds at all -- some phonemes have a completely
different physical sound, but are heard the same way, depending on
context, while sometimes the exact same sound in different contexts
is heard as two different phonemes -- hence one can hear a physically
identical sound as a different psychological "sound" in the context
of language. This is one of the reasons it's taken so long to develop
voice recognition software!

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> Yes, absolutely.

Fascinating. That would also mean that our perceptual measuring of an
interval by ear would most definitely rely on either this "contextual"
hearing or placing it in an intonational context (such as locking in a
just interval) rather than being able to tune a pitch x number of
cents away from some other given pitch. What I find most interesting
is that I usually considered pitch as one of the (more or less)
absolutes, and that context was more implicit in other areas of music.
If we're saying pitch is contextual as well, then it almost makes a
complete picture of music being completely plastic. Which I consider a
plus, as opposed to black-and-white.

Cheers,
Jon

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > Yes, absolutely.
>
> Fascinating. That would also mean that our perceptual measuring of
an
> interval by ear would most definitely rely on either
this "contextual"
> hearing or placing it in an intonational context (such as locking
in a
> just interval) rather than being able to tune a pitch x number of
> cents away from some other given pitch.

Perhaps, though I think the point in this discussion was that the
main way context makes two enharmonically equivalent intervals sound
different is *apart* from the impression of how wide they are or how
many cents they span -- rather, they sound different along a
different dimension altogether. A dimension in the conceptual space
in which the "tonal" (diatonic or whatever) meaning of the music
plays itself out. When one listens to music, one is supposedly
experiencing events in *this* space -- the categorization of all
intervals in terms of their sizes in cents is something that requires
one to direct one's attention away from this sphere, and toward the
less "musical", more "measurement" sphere of absolute-interval-size
judgments.

> What I find most interesting
> is that I usually considered pitch as one of the (more or less)
> absolutes,

Aha -- well then, you still have some interesting things to learn,
notwithstanding the above. There are many contexts in which one can
give the impression of pitch changing while the frequency actually
stays the same. One of these is by changing the loudness -- pitch
appears to shift. Another way is by exploiting the tendency of pure
tones to "push" one another "apart" in pitch, subjectively speaking.
So context *can* affect the perceived size of a given interval.
Terhardt's pages discuss some of these effects, but they seem to be
down at the moment.

> and that context was more implicit in other areas of music.
> If we're saying pitch is contextual as well, then it almost makes a
> complete picture of music being completely plastic. Which I
consider a
> plus, as opposed to black-and-white.

For what it's worth, few things in the visual realm (sizes, shapes,
colors) are immune from context-dependence either. I'm sure you've
seen at least a few of the "optical illusions" intended to illustrate
this point. Does this make art or music completely plastic? I'd say
it just makes them human endeavors.

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:

> Fascinating. That would also mean that our perceptual measuring of an
> interval by ear would most definitely rely on either this "contextual"
> hearing or placing it in an intonational context (such as locking in a
> just interval) rather than being able to tune a pitch x number of
> cents away from some other given pitch.

Functional harmony rules.

> If we're saying pitch is contextual as well, then it almost makes a
> complete picture of music being completely plastic.

Not hardly. There's a difference between a sweet third and a sharp
third even when functionally they are the same.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> Aha -- well then, you still have some interesting things to learn...

No way.

> There are many contexts in which one can
> give the impression of pitch changing while the frequency actually
> stays the same.

"Barber-pole" effects as well, I imagine.

> So context *can* affect the perceived size of a given interval.

Your examples seem to show a broader use for the term 'context' than I
was thinking, but no matter, I see where this goes.

> For what it's worth, few things in the visual realm (sizes, shapes,
> colors) are immune from context-dependence either. I'm sure you've
> seen at least a few of the "optical illusions" intended to illustrate
> this point.

Yes, I actually had this in mind: where in the music example the exact
same interval distance sounds different depending on what surrounds it
or how it is approached and departed; this would seem similar to the
visual things that say "which color is darker", where the same color
appears different in the context of it's surroundings.

> Does this make art or music completely plastic? I'd say
> it just makes them human endeavors.

I meant plastic as in "malleable", and in that sense it is more human
(to me).

Cheers,
Jon

Hi Paul,

On Tue, 25 Oct 2005, "wallyesterpaulrus" wrote:
> > "Sharpness", as represented by accidentals, is just
> > a notational device for adjusting the pitch of the note,
> > that is for most instruments, the frequency of the
> > fundamental component of the note. In this way it
> > does not differ at all from the device of placing notes
> > at different heights on a staff. We can readily see
> > this must be so, because, when playing an instrument
> > of fixed intonation, such as a piano or organ, if that
> > instrument is tuned to 12-EDO, a player must play
> > exactly the same note (strike exactly the same key)
> > when the notation says Cx as when it says D or Ebb.
>
> And yet some composers do carefully observe these distinctions in
> notation.

Yep; I'm one of them. I see no sense in notating an F#
in Dmajor as Gb - it would only confuse the player.
Besides, it's inefficient, as I would then need to use
another accidental to indicate that a G natural in that
key is, indeed, natural.

> Some theorists argue that these distinctions are actually
> *heard*, because of context, ...

I'd say, as I have said before, that we take the 12-EDO
note as a workable approximation to the ideal note,
whether F# or Gb, that belongs to whatever idealised
version of the scale we have in mind (whether consciously
or not). So these notes are not actually *heard*, so much
as *imagined*.

So often we have read theorists in the past decrying the
"barbarism" and "crudity" of 12-EDO tuning - Ravi Shankar
springs to mind. Perhaps, on the contrary, those of us
(many millions, I might add) who can imagine that they are
hearing the "right" sounds when presented with some poor
approximation, are instead more sophisticated than we
know ... ;-)

> ... and say it's naive to think ...

That's me, I'm a *very* na�ve thinker!

> ... that we hear
> each note on the instrument as the same "grammatical" or "syntactic"
> entity in the music even when it's spelled differently -- just as we
> hear an augmented second as a very different interval from a minor
> third, even though *physically*, absent context and musical meaning,
> they are the same on a piano or organ.

Of course context makes all the difference!

I think (perhaps na�vely?) that we actually hear much less than
we think we do; we imagine the rest. Or that "hearing" is another
of those words, like "seeing" and "reading", that seems, on the
face of it, to concern itself with the act of sensing, whereas in
reality sensing is just the tip of the iceberg; what the brain does
with the sensory stimuli is many times more important AND more
variable than the sensing itself.

Academic, philosophic & psychological *tomes* have been written
on the nature of perception, apperception and the like; the
take-home message is that hearing, seeing etc are complex
activities that we have to learn to do, and that everyone will learn
to do differently.

Regards,
Yahya

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Hi Linas,

On Tue, 25 Oct 2005 Linas Plankis wrote:
>
> Hi, Yahya,
> i continue answering to your questions:
> >> ... other interesting possibilities this rule has
> >> the consequence, that deviation of the pitch
> >> frequencies of a given scale relative to an equally
> >> tempered scale is the main characteristic of a given
> >> scale (meaning, that tunings like 12-EDO are more sets
> >> of necessary scales than musical scales themselves).
> >> And it works.
> >
> >??? How?

You don't have an answer for this?

> >> ... After making this conclusion we could
> >> even skip the dimension x and use the pitch frequency
> >> only, but already knowing the rule.
> >
> >OK, suppose we skip the second dimension, and use only
> >the fundamental frequency of a note. (This is what the
> >rest of us still do.) What does "knowing the rule" mean?
> >That somehow the (squared, say) deviations of the
> >tuning from 12-EDO are useful harmonically? I think I
> >showed above that they are not.
>
> Perhaps you should draw a graph of deviations for
> better visuality, but no one may insist on it. You
> would see three moments, that are very obvious:
> 1.Graph has a number of peaks.
> 2.Steps of growing are different in size from steps of
> descending.
> 3.And, without any doubt, the graph will have a
> certain number of points.

Easy enough to do; I draw graphs in my head! Unless
I need to take measurements from them.

> These characteristics are essential for a scale. And
> some secondary characteristics, as eg number of points
> / number of peaks, also could be successfully used.
>
> Classifying is easy then. For example, we can take
> scales, that have 17 points (are 17-tonics) and 3
> peaks, as a group.

Sure, I get it: you can classify scales by any feature
you like of the graphs of their deviations from some
norm, whether that be 12-EDO, JI diatonic heptatonic,
or whatever seems appropriate. What's more, you are
free to choose which features you want to compare
and contrast. This is a method you are proposing by
which we can compare or classify tunings. What I
*still* don't get is why you believe these classifications
have any *musical* significance.

> Also there's an interesting feature of scales. Some of
> them have growing steps smaller than descending steps,
> but others, in contrary, have growing steps bigger
> than descending ones (compare diatonic and enharmonic
> scales by Boetius / Aristoxenes). This feature is also
> useful classifying scales.

Not entirely sure of your meaning here ... perhaps
something was "lost in translation"?

> How it works? Perhaps You should try: Create two
> scales with similar or, better, same, characteristics
> of the list above, except one. Perhaps it would be
> better argument than all my arguing here.

While I could do as you suggest, I have no guarantee
that my results will take me in the kind of direction
you intend. Also, I don't have unlimited time for such
blind experimenting. Finally, I am asking you these
questions in hopes of getting answers that other
members of this list may find useful, not just me.

Perhaps it would be better if *you* were to create
two such scales, then present us with two variants
of a piece of music, using first one scale then the
other. This would give you the opportunity of showing
us just how your proposed method will change the
music we might make. And then we could judge for
ourselves whether or not the method was successful.

Consider this a challenge! :-)

> The short answers:
>
> >Indeed. When you tune the piano's 88 notes, you
> >create a _gamut_ from which you can draw 2^88
> >different musical _scales_ for purposes of composition,
> >not all equally useful.
>
> Looking at the common practice period, one can see,
> that the scale with 12 keys was used as a scale for
> the heptatonics. This understanding is possible,
> concerning not only the common practice period. I
> simply exclude this taking of a super-scales for what
> we speak about. And it's quite clear thing, so i take
> this your idea more as a joke.

Linas, when I make a joke, I insert the following
smiley: :-)

But I was not joking when I wrote the previous
sentence! Nor was I joking when I wrote about the
2^88 scales on the piano. If you choose to restrict
your enquiries to octave-equivalent scales, then indeed
you have a gamut of a mere 12 notes, from which you
may draw 2^12 scales. And if you restrict things even
further, to scale classes, in which one major scale is as
good as any other, then you have only 2^12/12 scale
classes to worry about. Or you could look at the
"common practice period" and end up with perhaps
just *three* scale classes.

However, if your intention is to offer methods that
musicians may use to *expand* their musical resources,
why would you ever want to restrict the field of play
so utterly? Beats me ...

> >> 2.It could be used for classifying as we've discussed.
> >I'm not clear what you mean here.
> If the answer above isn't sufficient, the answer
> still stands in the queue.

Yes, I see what you mean by classifying. But can you
demonstrate that these classifications are useful?

> >> ... you should
> >> notice, that i simply used this. I divided the rest
> >> from the fundamental frequency.
> >> Now, why the number of dimensions is 2. An adding one
> >> dimension may appear helpful, but if we added more, it
> >> would cause too complicated system , as I think.
> >
> >Not if that represented the nature of musical
> >reality better. Would our description of space-time
> >be better if we used only, say, time and radial distance?
> >Although that's perfectly adequate for some purposes
> >(eg "as the crow flies"), it is almost useless when we
> >need to describe the motions of, say, helicopters.
> >Essential complexity is not a thing to be avoided. If
> >an adequate description of phenomena requires 17
> >dimensions, then that's exactly how many we should
> >use. Otherwise, we will simply get the wrong results,
> >or no results at all.
>
> If we discussed how to synthesize sound in generally,
> i should agree with You. The same is about
> synthesizing timbres for electronic instruments. But
> when we speak about pitch, there are some limits. We
> have our notational system at most two-dimensional,
> and not 3- or more-dimensional, and it commits us to
> be careful, especially when starting.

Take a deep breath, Linas! :-)

/*begin rant*/
I've been designing software and business systems for
many years now, with some success. A musical notation
system is a business system for musicians to use, so I
think my experience may have some relevance here.

What I know beyond a shadow of doubt is this truism,
that's in all the textbooks on system and software
design, but that everyone seems hell-bent on proving
for themselves the hard way:
You've got to get your analysis right
*before*
you start coding.

The short form of this rule is pretty much: GIGO
("Garbage In, Garbage Out").

And yes, I do know about being careful. A single
mistake when starting out can cost you very much more
time and trouble later on, or even make the system you
build completely unworkable; useless.

So I exhort you to get ALL the dimensions you think
exist down on paper - even if there ARE 17 of them!
I know you've been having a lot of trouble convincing
me that there really is a second dimension to tuning,
but if you can do that, then you can convince me of all
the others too.

Leaving any out now will only make things harder later.
Do you know, I think that you overrate the difficulty
of designing a musical notation system to handle
several dimensions - it's been done already with great
success. The "common practice period" notation could
readily handle these independent dimensions with
sufficient accuracy for the needs of the period:
- time
- pitch
- dynamics
- articulation

As modern composers continue to push boundaries, of
course it will be a challenge to design a suitable notation
system to meet their needs. Some interesting and
valuable experiments have already been made, for
example the Sagittal and HEWM notations (George
Secor, Dave Keenan, and Joe Monzo can tell you more).

But the most important step in designing a new notation
system remains this: discovering exactly what it must
notate. Without this analysis, the result will fail to meet
the needs it was supposed to.
/*end rant*/

> >Very intently indeed. I though[t] I understood that -
> >at least in your extended example in the abstract -
> >you identifed the second dimension as the harmonic
> >distance (measured in steps along the circle of fifths)
> >between two scale degrees.
>
> The second dimension, as I understand it, isn't a
> purely mathematical thing.

Well, I have a little difficulty with this concept.
Without insisting (like Aristotle would have) that a thing
either is mathematical or is not mathematical, I'd like
to say that you *are* presenting this idea of yours using
the language of mathematical concepts - such as
"dimension" and "frequency". Therefore it's rather hard
to think that it's not mathematical! Surely you are saying
at least that some aspects of it are mathematical? If this
is true, what are they?

Or is it all pure allegory, in which you use the language of
number to confuse and mystify us, like any fairground
numerologist? :-) No, I don't think you're deliberately
confusing us - but you're still confusing me, for one.

> If it was a result from the
> formula only, what are we discussing about? I simply
> would have to say, that �There's a formula, that helps
> much in tuning. Let's try it�. But how the formula
> works, anybody may try explain differently from me
> too.

Are you saying we can all interpret this any way we like?
8-0

> E. g. like you say:
>
> >One of the other list members implied recently that
> >we experience inflection - that is, performance
> >deviations from the basic scale, eg a JI diatonic scale -
> >in a way that differs from our perceptions of the scale
> >itself. I think there may be something to this.
>
> However i don't think, that men hear deviations from
> an (ET in our case) scale. I suppose, that the second
> dimension should be proportional (taking logarithmic
> values eg in cents) to the deviation, when we want to
> hear specifically musical sounds.

Hmmm ... I think the inflections I hear are log deviations
from a JI diatonic 7 note scale, and also their time
envelopes. I'm pretty sure my hearing isn't subtle or quick
enough to pick up deviations from a 12-EDO chromatic
scale, at least at normal musical speeds.

Regards,
Yahya

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On Mon, 24 Oct 2005 "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > I'm not really interested in what it's "possible to prove".
> > Just call me a Doubting Thomas, and SHOW me! :-)
>
> What I feel is lacking so far are precise definitions. There isn't
> much point in postulating a second dimension if you cannot give a
> clear statement of what it is.

Gene, that was indeed my own first reaction, since
I was interpreting what *looked* like maths as tho
it *were* maths.

But I'm increasingly convinced that Linas is using
this "second dimension" concept more as an allegory
than as maths. For example, in our last exchange,
he was emphasising how we might use the (log)
deviations of notes from some standard scale as
a means of classifying or categorising tunings into
a whole number of discrete classes. It seems that
the class, or pattern made by these deviations, is
more important to him than any more subtle or
continuous measure based on them, eg the RMS
deviation. BTW, this notion of tuning class is not
entirely foreign to historical usage - we have well
temperaments, meantones, _temp�raments ordinaires_
and so on.

There are obvious advantages to distinguishing
classes of tunings and temperaments; chiefly brevity
of reference, but perhaps also they give us a feeling
for the "flavour" of the tuning before even hearing it.

Anyway, I'm trying (struggling?) to keep an open
mind, so that we don't miss anything of value that
Linas may be trying to say. All that I'm sure of so
far, is that Linas' English is *much* better than my
Estonian! :-)

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:

> Yes, I actually had this in mind: where in the music example the
exact
> same interval distance sounds different depending on what surrounds
it
> or how it is approached and departed;

This was another sort of effect that we've observed on this list.
We've had extremely accomplished, experienced non-microtonal
musicians come here, listen to certain microtonal examples, and state
with conviction that they heard certain intervals . . . but when the
example was played backwards, they heard something quite different
than just a backwards rendition of these same intervals. The effect
seemed closely tied to a lifetime of conditioning with non-microtonal
music, and what is known as "categorical perception" -- something
that makes much microtonal music sound simply like out-of-tune music
to many listeners at first.

> this would seem similar to the
> visual things that say "which color is darker", where the same color
> appears different in the context of it's surroundings.

> > Does this make art or music completely plastic? I'd say
> > it just makes them human endeavors.
>
> I meant plastic as in "malleable", and in that sense it is more
human
> (to me).

If the composer could control these illusions (putting aside that
different people will hear them differently), then I would agree with
the "malleable" assessment. As it is, though, the composer really has
no control over them, they occur whether the composer wants them to
or not, so I'm not sure this is an indication of more "malleability"
in the art. But it's very much an indication that music is an
endeavor of human ears and imaginations, and attempts to systematize
it on paper are likely to fall short of their goals. (Music -- not
mere tuning :) )

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > However i don't think, that men hear deviations from
> > an (ET in our case) scale. I suppose, that the second
> > dimension should be proportional (taking logarithmic
> > values eg in cents) to the deviation, when we want to
> > hear specifically musical sounds.
>
> Hmmm ... I think the inflections I hear are log deviations
> from a JI diatonic 7 note scale,

I hope I'm not taking this out of context, but . . . when you hear
something in a major key, and the interval between the 2nd and 6th
notes in the scale is 700 cents, do you hear this interval as 20
cents too wide??

> and also their time
> envelopes. I'm pretty sure my hearing isn't subtle or quick
> enough to pick up deviations from a 12-EDO chromatic
> scale, at least at normal musical speeds.

If I've been listening to only 12-equal music for a while, then it's
certainly easier or at best no harder for me to pick up deviations
from a 12-EDO chromatic scale than from any other scale, melodically
speaking. Do you not find this?

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> The effect
> seemed closely tied to a lifetime of conditioning with non-microtonal
> music, and what is known as "categorical perception" -- something
> that makes much microtonal music sound simply like out-of-tune music
> to many listeners at first.

What's interesting is that listeners will clearly hear that
vertically, the sonorities are not harsh, but it will still sound out
of tune. It seems to be the melody line itself which sounds out of tune.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > The effect
> > seemed closely tied to a lifetime of conditioning with non-
microtonal
> > music, and what is known as "categorical perception" -- something
> > that makes much microtonal music sound simply like out-of-tune
music
> > to many listeners at first.
>
> What's interesting is that listeners will clearly hear that
> vertically, the sonorities are not harsh, but it will still sound
out
> of tune. It seems to be the melody line itself which sounds out of
tune.

Yes, and also certain harmonic intervals we've been conditioned to
expect -- I think that the ratios 19:1, 19:2, and 19:4 may be pretty
important in describing the way 12-equal-conditioned listeners hear
the interval between the bass "third" and the treble "root" of a
multi-octave or open-voiced first-inversion major triad . . . this
chord in 5-limit JI always sounds the most out-of-tune to me after a
long spell of only 12-equal listening. The root-position open-voiced
minor triad is next, probably because of the same 19th harmonic
approximation in 12-equal.

Hi Paul,

On Wed, 26 Oct 2005 "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > > However i don't think, that men hear deviations from
> > > an (ET in our case) scale. I suppose, that the second
> > > dimension should be proportional (taking logarithmic
> > > values eg in cents) to the deviation, when we want to
> > > hear specifically musical sounds.
> >
> > Hmmm ... I think the inflections I hear are log deviations
> > from a JI diatonic 7 note scale,
>
> I hope I'm not taking this out of context, but . . . when you hear
> something in a major key, and the interval between the 2nd and 6th
> notes in the scale is 700 cents, do you hear this interval as 20
> cents too wide??

I guess that when I've heard *enough* of some music
in a diatonic 7 note scale tuned in 12-EDO, I hear each
interval of 700 cents as an acceptable fifth.

> > and also their time
> > envelopes. I'm pretty sure my hearing isn't subtle or quick
> > enough to pick up deviations from a 12-EDO chromatic
> > scale, at least at normal musical speeds.
>
> If I've been listening to only 12-equal music for a while, then it's
> certainly easier or at best no harder for me to pick up deviations
> from a 12-EDO chromatic scale than from any other scale, melodically
> speaking. Do you not find this?

There are two factors at play here. One is the tuning,
which we have stipulated as 12-EDO. The other is the
scale or melody class - of which a convenient
representation for now will be the simple categories
pentatonic, heptatonic and chromatic. Does it use 5,
7 or 12 notes from the gamut?

In this 12-EDO situation, if the music is essentially 7-note
diatonic, and not especially chromatic, I think I hear any
chromatic inflection as a "colouring" of one of those 7
scale notes. I'm fairly sure I don't mentally compare the
inflection to an internalised 12-EDO ideal. As proof, when
playing the violin unaccompanied, I would invariably move
the inflection in the direction of its resolution, usually the
following note. For example, in playing the following figure
in Aminor:

| B A G# A | A G# F# G#| A

the first G# would be sharper than written, while the
second would be "more equal" - more like the written note.
I don't know if this is unusual, but it certainly sounds
more musical to me than playing strict 12-EDO. I suspect
(but have no proof) that much of the colour of small
ensemble playing arises from this kind of pushing around
of the intonation.

Similarly, if the music is essentially pentatonic, I find
any note foreign to that pentatonic scale to be less
critical in inflection than the scale notes. Tho I can't
recall playing much pentatonic stuff on the violin (and
haven't had one for over 10 years), I suspect I would
also push those foreign notes around a bit.

Hope this answers your question.

Regards,
Yahya

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--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I think that the ratios 19:1, 19:2, and 19:4 may be pretty
> important in describing the way 12-equal-conditioned listeners hear
> the interval between the bass "third" and the treble "root"

Whoops . . . I meant treble "fifth", of course . . . :)

Hi, Yahya.

>Well, I have a little difficulty with this concept.
>Without insisting (like Aristotle would have) that a
thing
>either is mathematical or is not mathematical, I'd
like
>to say that you *are* presenting this idea of yours
using
>the language of mathematical concepts - such as
>"dimension" and "frequency". Therefore it's rather
hard
>to think that it's not mathematical! Surely you are
saying
>at least that some aspects of it are mathematical?
If this
>is true, what are they?
I use mathematical language, and it's so, but it
doesn't mean, that all is math. I meant, that the
second dimension isn't a purely mathematical
consequence (got for instance from formulas as their
result) of some calculations. If it were so, it was no
need to introduce the second dimension at all. It's
simply very difficult to imagine, that the thing, that
we got used to imagine one-dimensional appears a bit
wider. The most of us, as I think, simply use pitch as
a line in a plane. Ones do with a more straight line,
others with more curved; that's why an understanding
of pitch has so many interpretations.
Perhaps the rule of harmonic tuning appears more
mathematical thing than others (the same way like the
Pythagorean rule of multiple frequencies), but if the
analysis of deviations isn't interesting for us, we
simply could skip all mathematics, taking the formula
of the result.
>Hmmm ... I think the inflections I hear are log
deviations
>from a JI diatonic 7 note scale, and also their time
>envelopes. I'm pretty sure my hearing isn't subtle
or quick
>enough to pick up deviations from a 12-EDO chromatic
>scale, at least at normal musical speeds.
You can hear inflections from any scale, whose
intervals aren't too small, if You are trained. But i
doubt, if this hearing is automatic. That's what i
mean. Although there is no doubt, that our inner meter
measures these deviations in logarithmic measures.
This fact inspires us to look further. It's because,
if we can know, what kind of measures uses such
subjective and unmeasurable processes, perhaps we can
know more too?
>But I'm increasingly convinced that Linas is using
>this "second dimension" concept more as an allegory
>than as maths. For example, in our last exchange,
>he was emphasising how we might use the (log)
>deviations of notes from some standard scale as
>a means of classifying or categorising tunings into
>a whole number of discrete classes. It seems that
>the class, or pattern made by these deviations, is
>more important to him than any more subtle or
>continuous measure based on them, eg the RMS
>deviation. BTW, this notion of tuning class is not
>entirely foreign to historical usage - we have well
>temperaments, meantones, _temp�raments ordinaires_
>and so on.
Not allegory, but the interpreting of the second
dimension as of result of the deviations formula is
wrong. If we have a proper interval of x dimension, a
sound may have, according the theory, any value from
this interval, with no prescriptions. And in the case
only, when we want to put a sound to a certain tuning
scale, we give the values, that are got from the
formula, for the x dimension. This formula is for a
giving order to a tuning, not for a defining the
second dimension.
I can give an easy example. Say, we use C major,
having tuned second dimension values for it. Let also
fundamental frequencies of sounds in our tuning be
from 12-ET for easier comparing. Now, let's have to
change a tuning from C major to G major. In this case
we should change fundamental frequency once (F -->
F#), but the second dimension is different thing. If
we change a scale, we should retune each value of the
previous scale, putting x dimension value of note C to
note G , etc. etc. till F# gets the value, that
depended to B before it.
Linas

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Hello Carl and Yahya,

>I agree that MIDI music could benefit from more data
>than standard MIDI files contain. But the MIDI
>standard seems fairly extensible, if one were to
write
>a custom synthesis system.
>
>Do you have any musical examples (audio or scores)?
>
>-Carl
>

I'd like to upload two mp3 files, but they've appeared
too big (2x3.5MB) for disk space of the yahoo group. I
think I may send you these files personally?

>> But if anybody wants to try possibilities of the
>> adding the x-parameter to real tuning, there is a
>> restriction. Such experiments aren't possible,
using
>> using software that works exclusively with midi
data,
>> nor with any EMI-s or other devices, that have
>> predefined one-dimensional tuning scale. Such
>> experiments are possible with the synthesizing
>> programs like csound or clm.
>
>Suppose I use a MIDI-capable compositional tool,
>such as NoteWorthy Composer. With this, I create
>MIDI files, which I then use to play my MIDI
>keyboard. That keyboard incorporates a whole pile of
>sampled instruments in its hardware. In the MIDI
file,
>I include pitch-bends as well as standard MIDI note
>numbers, so that I can sweep through all frequencies
>from the lowest MIDI note to the highest. Because
>the keyboard has a high degree of polyphony, I can
>layer many instruments simultaneously.
>
>Now suppose I use a soft synth like csound, or even
>a hardware synth such as my old Moog. I will grant
>you that these synths can create some fairly simple
>sounds that I can't get from the MIDI keyboard.
>But almost any of the more complex sounds they
>make can be approximated on the MIDI keyboard.
>This is as true of their pitch as of their timbre.

I'm not sure, that one may get everything from MIDI
including both pitch and timbre. But it's true, that
MIDI is very flexible thing. In the case, You are
right, the only advantage of Csound and similar
programs would be, that it doesn't predefines way of
thinking. It's hard job to get anything more musical
using them, but it's a free making of sound. A
composer can feel himself like a painter, and this
feeling is great, perhaps it's a sufficient
compensation for few times bigger job, needed for this
kind of software.

Linas

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Hi Linas,

> I'd like to upload two mp3 files, but they've appeared
> too big (2x3.5MB) for disk space of the yahoo group. I
> think I may send you these files personally?

It's best if you have your own web server to use,
and post URLs to the files there.
Otherwise, you may try to send them to me at
clumma@gmail.com.

Thanks!

-Carl

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> As modern composers continue to push boundaries, of
> course it will be a challenge to design a suitable notation
> system to meet their needs. Some interesting and
> valuable experiments have already been made, for
> example the Sagittal and HEWM notations (George
> Secor, Dave Keenan, and Joe Monzo can tell you more).

See
http://dkeenan.com/sagittal/
http://tonalsoft.com/enc/h/hewm.aspx

-- Dave Keenan

Hi.

I uploaded the files at
http://geocities.yahoo.com/linasrim/GM-odan.zip
These are two examples (zipped in one file) of my
experiments. The both examples are almost identical
(It's an old Lutheran singing music in two different
tunings). Perhaps sounds aren't elaborate very much in
the both examples, but my conception is used in the
both. I did them with Csound (except, naturally, the
final conversion to mp3 and zip). It's up to you, to
try to guess, if fundamental frequencies of sounds are
identical in both examples or not.

Linas

--- Carl Lumma <clumma@yahoo.com> wrote:

> Hi Linas,
>
> > I'd like to upload two mp3 files, but they've
> appeared
> > too big (2x3.5MB) for disk space of the yahoo
> group. I
> > think I may send you these files personally?
>
> It's best if you have your own web server to use,
> and post URLs to the files there.
> Otherwise, you may try to send them to me at
> clumma@gmail.com.
>
> Thanks!
>
> -Carl
>
>
>
>
>

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--- Linas Plankis <linasrim@yahoo.com> wrote:

>
> I uploaded the files at
> http://geocities.yahoo.com/linasrim/GM-odan.zip

Sorry! The address was wrong. It should be:
http://www.geocities.com/linasrim/GM-odan.zip

Linas

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--- Linas Plankis <linasrim@yahoo.com> wrote:

>
> I uploaded the files at
> http://geocities.yahoo.com/linasrim/GM-odan.zip

Sorry! I can only explain this knowing what day is
today :). The true address is:

http://www.geocities.com/linasrim/Gm-Odan.zip

once more sorry.

Linas

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> > I uploaded the files at
> > http://geocities.yahoo.com/linasrim/GM-odan.zip
>
> Sorry! I can only explain this knowing what day is
> today :). The true address is:
>
> http://www.geocities.com/linasrim/Gm-Odan.zip
>
> once more sorry.

Success!!

-Carl

> > > I uploaded the files at
> > > http://geocities.yahoo.com/linasrim/GM-odan.zip
> >
> > Sorry! I can only explain this knowing what day is
> > today :). The true address is:
> >
> > http://www.geocities.com/linasrim/Gm-Odan.zip
> >
> > once more sorry.
>
> Success!!

They both (Gm-Odan-74x-1.mp3 and Gm-Odan-74x-rom-1.mp3)
sound good. What's the difference?

What tuning was used?

-Carl

Hi.

> >
> > Success!!
>
> They both (Gm-Odan-74x-1.mp3 and
> Gm-Odan-74x-rom-1.mp3)
> sound good. What's the difference?
>
> What tuning was used?
>
> -Carl
>

The tuning is:
(1) Fundamental frequencies in the both cases are:
G : 1
Ab : 17/16
A : 9/8 (440 Hz)
Bb : 6/5
B (Cb): 5/4
C : 4/3
Db : 7/5
D : 3/2
Eb : 8/5
E : 5/3
F : 9/5
Gb (F#) : 15/8
(2) Parameters of micro-vibrato and duration of echo
(ie of the longer reverberation) are regulated,
proportionally to value, get from the rule of harmonic
tuning from my abstract
(http://www.geocities.com/linasrim/2-dtuning.pdf ),
compensating the second dimension this way.
The file �Gm-Odan-74x-1.mp3� has an enneatonic tuning,
ie a tuning, that has nine changes of level of the
2-nd dimension per octave:
G change Ab ch. A ch. [Bb, B] ch. C ch. Db ch. D ch.
[Eb, E] ch. [F, F#] ch. G
The file �Gm-Odan-74x-rom-1.mp3� has a heptatonic
tuning, which one has 7 changes of 2-nd dimension
level per octave:
[Gb, G] change [Ab, A] ch. Bb ch. [Cb, C] ch. [Db, D]
ch. [Eb, E] ch. F ch. [Gb, G]
These are the main points of the tuning, i think.

Linas

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> > > Success!!
> >
> > They both (Gm-Odan-74x-1.mp3 and
> > Gm-Odan-74x-rom-1.mp3)
> > sound good. What's the difference?
> >
> > What tuning was used?
> >
> > -Carl
>
> The tuning is:
> (1) Fundamental frequencies in the both cases are:
> G : 1
> Ab : 17/16
> A : 9/8 (440 Hz)
> Bb : 6/5
> B (Cb): 5/4
> C : 4/3
> Db : 7/5
> D : 3/2
> Eb : 8/5
> E : 5/3
> F : 9/5
> Gb (F#) : 15/8
> (2) Parameters of micro-vibrato and duration of echo
> (ie of the longer reverberation) are regulated,
> proportionally to value, get from the rule of harmonic
> tuning from my abstract
> (http://www.geocities.com/linasrim/2-dtuning.pdf ),
> compensating the second dimension this way.
> The file "Gm-Odan-74x-1.mp3" has an enneatonic tuning,
> ie a tuning, that has nine changes of level of the
> 2-nd dimension per octave:
> G change Ab ch. A ch. [Bb, B] ch. C ch. Db ch. D ch.
> [Eb, E] ch. [F, F#] ch. G
> The file "Gm-Odan-74x-rom-1.mp3" has a heptatonic
> tuning, which one has 7 changes of 2-nd dimension
> level per octave:
> [Gb, G] change [Ab, A] ch. Bb ch. [Cb, C] ch. [Db, D]
> ch. [Eb, E] ch. F ch. [Gb, G]
> These are the main points of the tuning, i think.
>
> Linas

Aha! The light is beginning to dawn. Are you familiar
with Henry Cowell's _New Musical Resources_?

-Carl

Hi.
--- Carl Lumma <clumma@yahoo.com> wrote:

>
> Aha! The light is beginning to dawn. Are you
> familiar
> with Henry Cowell's _New Musical Resources_?
>
> -Carl
>
>
>
>
Ok. I know almost nothing about this Cowell's work.
One of my ideas, writing "the 2-d tuning", was to find
some similar ideas and persons, who can share them.
Thanks. (And i would be glad to get more information
about).

Linas.

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> > Aha! The light is beginning to dawn. Are you
> > familiar with Henry Cowell's _New Musical Resources_?
> >
> Ok. I know almost nothing about this Cowell's work.
> One of my ideas, writing "the 2-d tuning", was to find
> some similar ideas and persons, who can share them.
> Thanks. (And i would be glad to get more information
> about).

One of the things Cowell suggests is that not only can
scales be taken from the harmonic series, but rhythms,
dynamics, and other musical things as well. It struck
me that this is like what you were doing with vibrato.

-Carl

> > Ok. I know almost nothing about this Cowell's
> work.
> > One of my ideas, writing "the 2-d tuning", was to
> find
> > some similar ideas and persons, who can share
> them.
> > Thanks. (And i would be glad to get more
> information
> > about).
>
> One of the things Cowell suggests is that not only
> can
> scales be taken from the harmonic series, but
> rhythms,
> dynamics, and other musical things as well. It
> struck
> me that this is like what you were doing with
> vibrato.
>
One essential correction. I suppose that our hearing
of pitch is flexible at some extent. What means, that
if one makes some inessential changes of the waveshape
we deal with, we sometimes could hear it as change of
pitch, even when looking strictly the change was of
timbre. But the change should be lesser than ones,
which we consider as clearly different thing than
pitch (as tremolo, vibrato, reverberation etc.).
That's why I used words *micro*timbre, *micro*vibrato.

But conception of Cowell concerns more proportionating
pitch with rhythm and other parameters of music,
what's up to a composer. My ideas in �the
2-dimensional tuning� are more about tuning and they
don't concern how a composer deals with rhythm,
dynamics and such other parametric aspects of music.
Or, speaking more exactly, they concern all these
things, but no more than a selection between one or
another, say, violin affects an interpreting of a
composition. So, I speak about still a timbre, still a
vibrato, still a reverberation (and anyone could
extend it to �still a rhythm�, if he wants), but only
when a timbre, a vibrato ... have lost their
stand-alone musical significance and have been soaked
to the field of pitch. Actually nobody (or perhaps
almost nobody, statistically nobody) even hears them
as timbre, vibrato, ... although acoustically they
remain to be in this quality even unheard.

All these problems of limits and boundaries are
interesting both in mathematics and in music. They
often give unexpected and interesting results. And I
think, the things, what I speak about, more concern
namely these ideas of boundaries than what You say
about. But surely I don't say that ideas of Cowell
and �two-dimensionality� couldn't have anything in
common.

Linas

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> > One of the things Cowell suggests is that not only
> > can scales be taken from the harmonic series, but
> > rhythms, dynamics, and other musical things as well.
> > It struck me that this is like what you were doing
> > with vibrato.
> >
> One essential correction. I suppose that our hearing
> of pitch is flexible at some extent. What means, that
> if one makes some inessential changes of the waveshape
> we deal with, we sometimes could hear it as change of
> pitch, even when looking strictly the change was of
> timbre. But the change should be lesser than ones,
> which we consider as clearly different thing than
> pitch (as tremolo, vibrato, reverberation etc.).
> That's why I used words *micro*timbre, *micro*vibrato.

Ah. So this is tying timbre to tuning. Are you
familiar with the work of Bill Sethares?

http://eceserv0.ece.wisc.edu/~sethares/

He ties the spectral content of timbres to tunings,
and vice versa. Again, perhaps not the same as you
are doing, but maybe of interest.

-Carl

>
> Ah. So this is tying timbre to tuning. Are you
> familiar with the work of Bill Sethares?
>
> http://eceserv0.ece.wisc.edu/~sethares/
>
> He ties the spectral content of timbres to tunings,
> and vice versa. Again, perhaps not the same as you
> are doing, but maybe of interest.
>
> -Carl
>
>
>
>
Thanks, Carl.
I read the material by W. A. Sethares again intently
and with great interest. I'd been read it, if i
remember well, but only passing, for it is, as You say
rightly, not exactly the same as my idea.
What concerns timbre, I should agree with You, but we
should distinguish different cases here. Timbre is a
very wide thing. Well, all of us know, that it has
many frequential characteristics in it, even omitting
other, less regular characteristics. So, my ideas
cover mostly a part of timbre, that perhaps could be
called not timbre at all in a popular sense. For
instance, we have a definition of timbre in the help
glossary of the �Vienna 32� software

�Timbre differentiates a sound from another sound.
For example, it tells us the difference between the
sound of a violin and that of a tenor saxophone.
Timbre can be seen as the shape of a waveform. If the
shapes of two waveforms are different, their sounds
will differ. �

But we also should speak, additionally to it, yet
about a �grey zone� between these clearly heard timbre
changes and pitch (meaning fundamental frequency)
changes; so that timbre changes in this zone are heard
as spatial features of sound (micro-echo) or as a
complimentary effects to pitch, if they are heard at
all. So, my ideas concern mostly this �grey zone�, and
I'd like to avoid interference of tuning regularity to
the other part of timbre, because it warps traditional
sounding of instruments. But the warping of
traditional sounding already means a new aesthetics. I
think, that any new theory influences aesthetics in a
new way, despite one wants it or not, but I'd like to
reduce these innovations, preferring a description of
existing musical systems to a proclaiming of new
aestetical values.
What concerns Sethares , his compositions sound
unexpectedly, somewhere even excellent, but he changes
traditional sounding and the changes are connected
also with characteristic part of timbre (i e the part
of timbre, that allows us to differ sound of one kind
from sound of another). This position seems like a
declaring of new musical tradition. But I don't think,
that a new aesthetics is necessary to use capabilities
of the electronic music. On the other hand, Sethares
uses relations between timbre and pitch, and he is one
of the first, who did it more knowingly. Also his idea
to simulate consonances or dissonances is worth
knowing and deeper analyzing.

By the way, his and similar ideas affected my choice,
when I decided to speak about two-dimensionality of
pitch, but not about pitch-timbre relations (as
Sethares does). For we should speak about some
expansion of our pitch perception to the timbre zone,
but bot about necessary interrelating between the
timbre in general and pitch. Such an interrelating can
be used in order of experiment, but it isn't necessary
to call a sounds sequence musical, when we consider
characteristic timbre (not the one of the �grey
zone�).

It would be interesting to hear from others, what they
think about Sethares' experiments, both about the
aesthetic side of his compositions and about the ideas
in general.

Linas

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> For instance, we have a definition of timbre in the help
> glossary of the "Vienna 32" software
>
> "Timbre differentiates a sound from another sound.
> For example, it tells us the difference between the
> sound of a violin and that of a tenor saxophone.
> Timbre can be seen as the shape of a waveform. If the
> shapes of two waveforms are different, their sounds
> will differ."

Well, the waveform shape thing is wrong. You can change
it by changing the phases of partials but the timbre
won't change much.

> I'd like to avoid interference of tuning regularity to
> the other part of timbre,

You mean like Sethares?

> because it warps traditional
> sounding of instruments.

Sethares does some wild stuff, but warping the timbre
of a sound to fit, say, 19- or 22-tone equal temperament
won't change its timbre a whole lot. The Hammond organ
has partials very near to 12-tone ET.

> By the way, his and similar ideas affected my choice,
> when I decided to speak about two-dimensionality of
> pitch, but not about pitch-timbre relations (as
> Sethares does). For we should speak about some
> expansion of our pitch perception to the timbre zone,

I guess the question is: does the presence of microvibrato
change the percieved pitch of a tone?

> It would be interesting to hear from others, what they
> think about Sethares' experiments, both about the
> aesthetic side of his compositions and about the ideas
> in general.

I like his stuff plenty, but at this point I agree with
you -- it'd be nice to hear some more conventional-
sounding music from him (hi Bill!).

-Carl

--- Carl Lumma <clumma@yahoo.com> wrote:

>
> > I'd like to avoid interference of tuning
> regularity to
> > the other part of timbre,
>
> You mean like Sethares?
>
> > because it warps traditional
> > sounding of instruments.
>

I think, it's possible to make the presence of
tuning regularity in timbre even more obvious, than
Sethares does. The sentence was not directly about
Sethares, but that I'd prefer another way and why. It
all cames from the nature of sounds. We know, that
sound changes while traveling in the air. The most
changes are phasal shifts, but different fences met by
sound waves may result timbral changes too. So we are
acquainted to perceive some kind of timbre as possible
bias of space and not identify them with the source of
sound necessarily. So, while one synthesizes timbral
effects in this area, a listener could think, that
perhaps something is with space but not with a sound
source. When we overpass this area, listeners perceive
it as separate effects, as something form �another
stave�. By the way, some of the effects when timbre is
regulated in tuning, may be compared with the
situation when we have two or more co-tuned
instruments playing them sometimes in unison, but
often playing the higher notes according a
mathematical rule with one of them (which simulates
timbre in this case). But it's the same thing as if we
invented a mathematical rule for making chords, and
all depends on what the rule is. Some new rules may
cause some exotic sounding or introduce something like
a new aesthetics, i think. On the other hand, Sethares
also uses the nature of sounds to realize his idea, so
any contradiction, if it were, could be in the sense
of aesthetics only, but not in the sense of acoustics.

I think in this case, that perceived pitch isn't
singularly sharpness. We simply hardly distinguish
other properties of this perception in action,
although taking all this formally they can be
described as sharpness or height and some spatial
elements of sound perception. These spacial elements
could be encoded by microvibrato too. So, the
question, if the microvibrato affects perception of
pitch, should be answered by a question. What question
is: How we perceive pitch? It's comparable with how we
perceive coloured things. It's not easy to imagine any
object without a colour (when we consider that the
white is a colour too). The same way I've came to
conclusion, that the spatial elements of perception of
sound are unavoidable in it. And the sine wave is a
case like to the white colour among all colours. The
spatial elements, i speak about, have their place in
music without any doubt although they aren't perceived
clearly and notably like the height � sharpness �
fundamental frequency is.
How it all should be tuned and if the rules i propose
in �the 2-dimensional tuning� are adequate is a bit
different question. That's why i don't use acoustic
schemes in this abstract.

> Sethares does some wild stuff, but warping the
> timbre
> of a sound to fit, say, 19- or 22-tone equal
> temperament
> won't change its timbre a whole lot. The Hammond
> organ
> has partials very near to 12-tone ET.
>
> > By the way, his and similar ideas affected my
> choice,
> > when I decided to speak about two-dimensionality
> of
> > pitch, but not about pitch-timbre relations (as
> > Sethares does). For we should speak about some
> > expansion of our pitch perception to the timbre
> zone,
>
> I guess the question is: does the presence of
> microvibrato
> change the percieved pitch of a tone?
>

I think in this case, that perceived pitch isn't
singularly sharpness. We simply hardly distinguish
other properties of this perception in action,
although taking all this formally they can be
described as sharpness or height and some spatial
elements of sound perception. These spacial elements
could be encoded by microvibrato too. So, the
question, if the microvibrato affects perception of
pitch, should be answered by a question. What question
is: How we perceive pitch? It's comparable with how we
perceive coloured things. It's not easy to imagine any
object without a colour (when we consider that the
white is a colour too). The same way I've came to
conclusion, that the spatial elements of perception of
sound are unavoidable in it. And the sine wave is a
case like to the white colour among all colours. The
spatial elements, i speak about, have their place in
music without any doubt although they aren't perceived
clearly and notably like the height � sharpness �
fundamental frequency is.

How it all should be tuned and if the rules i propose
in �the 2-dimensional tuning� are adequate is a bit
different question. That's why i don't use acoustic
schemes in this abstract.

Linas

> > It would be interesting to hear from others, what
> they
> > think about Sethares' experiments, both about the
> > aesthetic side of his compositions and about the
> ideas
> > in general.
>
> I like his stuff plenty, but at this point I agree
> with
> you -- it'd be nice to hear some more conventional-
> sounding music from him (hi Bill!).
>
> -Carl
>
>
>
>

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> > I guess the question is: does the presence of
> > microvibrato change the percieved pitch of a tone?
>
> I think in this case, that perceived pitch isn't
> singularly sharpness.

Ah, in psychoacoustics, the word "pitch" is usually
taken to mean the singular property of height or
sharpness.

> We simply hardly distinguish
> other properties of this perception in action,
> although taking all this formally they can be
> described as sharpness or height and some spatial
> elements of sound perception. These spacial elements
> could be encoded by microvibrato too. So, the
> question, if the microvibrato affects perception of
> pitch, should be answered by a question. What question
> is: How we perceive pitch? It's comparable with how we
> perceive coloured things. It's not easy to imagine any
> object without a colour (when we consider that the
> white is a colour too).

But there are pitchless sounds. Many struck metal
objects produce tones that contain several equally
valid pitches. Different listeners may choose the
loudest, or the lowest of these as the pitch, or
be unable to decide. A single listener can also
'hear out' predominant components at will. Then,
there is white noise...

-Carl

--- Carl Lumma <clumma@yahoo.com> wrote:

> > > I guess the question is: does the presence of
> > > microvibrato change the percieved pitch of a
> tone?
> >
> > I think in this case, that perceived pitch isn't
> > singularly sharpness.
>
> Ah, in psychoacoustics, the word "pitch" is usually
> taken to mean the singular property of height or
> sharpness.
>
> > We simply hardly distinguish
> > other properties of this perception in action,
> > although taking all this formally they can be
> > described as sharpness or height and some spatial
> > elements of sound perception. These spacial
> elements
> > could be encoded by microvibrato too. So, the
> > question, if the microvibrato affects perception
> of
> > pitch, should be answered by a question. What
> question
> > is: How we perceive pitch? It's comparable with
> how we
> > perceive coloured things. It's not easy to imagine
> any
> > object without a colour (when we consider that the
> > white is a colour too).
>
> But there are pitchless sounds. Many struck metal
> objects produce tones that contain several equally
> valid pitches. Different listeners may choose the
> loudest, or the lowest of these as the pitch, or
> be unable to decide. A single listener can also
> 'hear out' predominant components at will. Then,
> there is white noise...
>
> -Carl
>
Perhaps sometimes I fail with English syntax. If we
use your way of thinking, the sentences have to be
put this way: The spatial elements of perception of
sound are unavoidable in it. It's comparable with how
one perceives coloured things. It's not easy to
imagine any object without a colour (the white is
considered a colour too). The same way the perceived
sharpness (or, in other words, the height), which is
the main side of pitch, is indivisible from these
spatial moments. (Or, in other words, we have the
expansion of our pitch perception to the timbre zone,
i spoke about). The most regular and the finest case,
when a sine wave is taken, plays a role similar to one
of the white colour in the colour perception. ___So,
I think, your question doesn't consider the thing that
i really wanted to say. But a possible answer to it
well suits with this allegory of colours. For one may
imagine something couloured (or white) without clear
contours easier than an explicit thing but absolutely
without a colour. This �something coloured (or white)
without clear contours� stands for cases when sounds
don't have an explicit pitch. In these cases spatial
sensation remains the only, and no musicians say, that
these sounds not have any smallest vestige of pitch at
all. Allusions between this �white� and �white� in
�the white noise� perhaps are accidental, although i
haven't thought about this similarity yet.

What You say about usage of �pitch� in
psichoacoustics, it's true without any doubt, and we
wouldn't have so many uncertainties in what we discuss
about here, if that weren't true. But, while i think
the sensations of sharpness and of micro-timbre are
indivisible, the explaining would be superformal
and quite long, if i tried to distinguish all it in
every even fiddly case. I think, it's a shorter way
here in the group, when i'm being asked about what is
unclear, rather than to write long texts, that
possibly were unclear too re my not good English.

But You aren't very precise too :). I think, the fact,
that drum sound is something like a very fast and
quite irregular vibrato, should be added to the
sentence about it. Otherwise it will seem like
description of few partials, that also are �tones that
contain several equally valid pitches� and �different
listeners may choose the loudest, or the lowest of
these as the pitch � (this is possible especially when
few *inharmonic* partials sound simultaneously or
listeners aren't musicians). __But this seeming not
preciseness is acceptable for me here, and i perhaps
don't seem to be very precise too.

Linas

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Hi Linas,

> What You say about usage of "pitch" in
> psichoacoustics, it's true without any doubt, and we
> wouldn't have so many uncertainties in what we discuss
> about here, if that weren't true. But, while i think
> the sensations of sharpness and of micro-timbre are
> indivisible, the explaining would be superformal
> and quite long, if i tried to distinguish all it in
> every even fiddly case. I think, it's a shorter way
> here in the group, when i'm being asked about what is
> unclear, rather than to write long texts, that
> possibly were unclear too re my not good English.
>
> But You aren't very precise too :). I think, the fact,
> that drum sound is something like a very fast and
> quite irregular vibrato, should be added to the
> sentence about it. Otherwise it will seem like
> description of few partials, that also are "tones that
> contain several equally valid pitches" and "different
> listeners may choose the loudest, or the lowest of
> these as the pitch " (this is possible especially when
> few *inharmonic* partials sound simultaneously or
> listeners aren't musicians). __But this seeming not
> preciseness is acceptable for me here, and i perhaps
> don't seem to be very precise too.

Yes, I think unfortunately we are up against a
language barrier at this point. But further audio
examples that illustrate your ideas are always
encouraged, and will admit to no such problems.

Cheers!

-Carl

> Yes, I think unfortunately we are up against a
> language barrier at this point. But further audio
> examples that illustrate your ideas are always
> encouraged, and will admit to no such problems.

Linas,

I'm listening again to your Gm-Odan-74x-rom-1.mp3
and Gm-Odan-74x-1.mp3 files. They sound very nice,
but I can't say I hear any difference at all
between them. And I notice they're exactly the
same size. Are you sure they're different?

-Carl

Hi, Carl.

The question:

I'm listening again to your Gm-Odan-74x-rom-1.mp3
and Gm-Odan-74x-1.mp3 files. They sound very nice,
but I can't say I hear any difference at all
between them. And I notice they're exactly the
same size. Are you sure they're different?

-Carl

is very natural.
Yes, the files are different, but they have very comparable values of encoding (a viewing using the hex mode in a file viewer shows it). They might be a case of equally sounding, but different things, some psychoacoustic isomorphisms, or how to call it, but actually the differences could be heard.
But it shows well the problem. The micro-timbral elements, i speak about, are heard very faintly, as some peripheral sensation, and it's not easy, to concentrate attention to them. The common level of them is heard better. Also an absence of change of some parameters is heard as something, that annoys with monotonic sounding. Perhaps only, the examples, i've uploaded, are too short, to hear this common level better. Naturally, that it's yet more difficult to hear differences among few pieces, that have sounding similar to equal.
I also have problem, that i don't realize, how much others can hear it more attentively, because my hearing is already trained on it during my experiments.
But i uploaded (to the tuning_files2) a longer piece for listening, without any comparative experiments. It's The Fantasy D minor for violin and piano (although performed with one virtual instrument, re using 0 type midi file for making a Csound score) K.397 by W. A. Mozart, the piece is performed with Csound, using a harp matrix from a sf2 file, with micro-timbral changes added (the variability of them is heptatonic). Enjoy it!
Linas

{ /tuning_files2 }

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Oh, sorry. I see, that the uploading is failed. Well, i'll try something yet.
Linas

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>> I'm listening again to your Gm-Odan-74x-rom-1.mp3
>> and Gm-Odan-74x-1.mp3 files. They sound very nice,
>> but I can't say I hear any difference at all
>> between them. And I notice they're exactly the
>> same size. Are you sure they're different?

> The micro-timbral elements, i speak about, are heard
> very faintly, as some peripheral sensation, and it's
> not easy, to concentrate attention to them. The common
> level of them is heard better. Also an absence of
> change of some parameters is heard as something, that
> annoys with monotonic sounding.

Ok, I'll listen for that.

> I also have problem, that i don't realize, how much
> others can hear it more attentively, because my hearing
> is already trained on it during my experiments.

Well, this list is a good place to have others listen.

> But i uploaded (to the tuning_files2) a longer piece
> for listening, without any comparative experiments. It's
> The Fantasy D minor for violin and piano (although
> performed with one virtual instrument, re using 0 type
> midi file for making a Csound score) K.397 by W. A. Mozart,
> the piece is performed with Csound, using a harp matrix
> from a sf2 file, with micro-timbral changes added (the
> variability of them is heptatonic). Enjoy it!

Thanks, I'm listening now (I got your offlist e-mail...
don't know why the tuning_files2 upload would fail...).
It's a great piece of music.

-Carl