Re: [tuning-math] question about quadratics

[Monz wrote:]
> Can someone tell me what application quadratic equations
> may have to determining string-lengths? I'm interested
> in possible applications for the purposes of determining
> both JIs and/or temperaments.

All I can think of is the C^2 = A^2 + B^2 theorem. You know, the
hypotenuse of a right triangle.

JdL

Hi, Joe! Long time no interact here. I don't remember the exact
formula for calculating string lengths, but it involves a constant
related to weight per unit length, then the longitudinal force
(tension on the string) and the length. I know once tension is fixed
the frequency is inversely proportional to the length, which is of
course a linear function of length and not quadratic. I believe the
relationship to tension is also linear, in this case directly
proportional. I don't recall quadratic equations ever entering the
picture.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Monz wrote:]
> > Can someone tell me what application quadratic equations
> > may have to determining string-lengths? I'm interested
> > in possible applications for the purposes of determining
> > both JIs and/or temperaments.
>
> All I can think of is the C^2 = A^2 + B^2 theorem. You know, the
> hypotenuse of a right triangle.
>
> JdL

--- In tuning@y..., BobWendell@t... wrote:
> Hi, Joe! Long time no interact here. I don't remember the exact
> formula for calculating string lengths, but it involves a constant
> related to weight per unit length, then the longitudinal force
> (tension on the string) and the length. I know once tension is
fixed
> the frequency is inversely proportional to the length, which is of
> course a linear function of length and not quadratic. I believe the
> relationship to tension is also linear, in this case directly
> proportional. I don't recall quadratic equations ever entering the
> picture.

J Gill:

From "Music, Physics, and Engineering", Olson, Dover, 1967, page 74:

Length (in cm) = (SQRT (T/M)) / (2F)
where T = tension (dynes), M = mass per unit length (grams),
and F = frequency (CPS).

> > [Monz wrote:]
> > > Can someone tell me what application quadratic equations
> > > may have to determining string-lengths? I'm interested
> > > in possible applications for the purposes of determining
> > > both JIs and/or temperaments.
> >
> > All I can think of is the C^2 = A^2 + B^2 theorem. You know, the
> > hypotenuse of a right triangle.

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
> > Hi, Joe! Long time no interact here. I don't remember the exact
> > formula for calculating string lengths, but it involves a
constant
> > related to weight per unit length, then the longitudinal force
> > (tension on the string) and the length. I know once tension is
> fixed
> > the frequency is inversely proportional to the length, which is
of
> > course a linear function of length and not quadratic. I believe
the
> > relationship to tension is also linear, in this case directly
> > proportional. I don't recall quadratic equations ever entering
the
> > picture.
>
> J Gill:
>
> From "Music, Physics, and Engineering", Olson, Dover, 1967, page
74:
>
> Length (in cm) = (SQRT (T/M)) / (2F)
> where T = tension (dynes), M = mass per unit length (grams),
> and F = frequency (CPS).
>
Bob:
Thanks! So I was wrong about the tension factor having a linear
relationship to frequency. Also mass is non-linearly related,
although inversely. For those who find it intuitively more
enlightening to see how frequency depends on string tension, length
and mass, this equation converts to solve for frequency as:

F = SQRT T/(2L * SQRT M)

This is therefore a quadratic equation after all, of the form:

4M (L SQRD * F SQRD) - T = 0

> > > [Monz wrote:]
> > > > Can someone tell me what application quadratic equations
> > > > may have to determining string-lengths? I'm interested
> > > > in possible applications for the purposes of determining
> > > > both JIs and/or temperaments.
> > >
> > > All I can think of is the C^2 = A^2 + B^2 theorem. You know,
the
> > > hypotenuse of a right triangle.

--- In tuning@y..., BobWendell@t... wrote:
> --- In tuning@y..., "J Gill" <JGill99@i...> wrote:
> > --- In tuning@y..., BobWendell@t... wrote:

> >
> > J Gill:
> >
> > From "Music, Physics, and Engineering", Olson, Dover, 1967, page
> 74:
> >
> > Length (in cm) = (SQRT (T/M)) / (2F)
> > where T = tension (dynes), M = mass per unit length (grams),
> > and F = frequency (CPS).
> >
> Bob:
> Thanks! So I was wrong about the tension factor having a linear
> relationship to frequency. Also mass is non-linearly related,
> although inversely. For those who find it intuitively more
> enlightening to see how frequency depends on string tension, length
> and mass, this equation converts to solve for frequency as:
>
> F = SQRT T/(2L * SQRT M)
>
> This is therefore a quadratic equation after all, of the form:
>
> 4M (L SQRD * F SQRD) - T = 0

J Gill:

Bob, I can see how you got to the step (2FL)*(SQRT(M))-SQRT(T)=0,
but the squares of individual terms are not algebraicly combinable,
as are the squares(or roots) of products (or ratios) of numbers.

A quadratic equation (canonical form): (A)x^2 + (B)x^1 + (C)x^0 = 0
with two(real or complex)roots at (-B [+or-] SQRT(B^2 - 4AC)) / (2A).

I have some knowledge of FM, and an excellent Textronix reference
book on modulation spectra entitled "Spectrum Analyzer Measurements;
Theory and Practice", by Morris Engelson, printed by Tektronix.

A mind reader I am not, however, and not a single post of
the "Uncertainty Principle" thread (including the original post)
clearly states the actual spectral situation to be modeled. I have
asked Paul a couple of times in this thread to clarify the situation
which he envisions, but I guess he missed those requests for info.

With a clear idea of what exactly is being talked about, I would be
interested in commenting (regarding the "Uncertainty thread, and FM
spectra of complex modulating spectrums), and hopefully assisting in
the process.

Best Regards, J Gill

Respectfully

>
>
>
> > > > [Monz wrote:]
> > > > > Can someone tell me what application quadratic equations
> > > > > may have to determining string-lengths? I'm interested
> > > > > in possible applications for the purposes of determining
> > > > > both JIs and/or temperaments.
> > > >
> > > > All I can think of is the C^2 = A^2 + B^2 theorem. You know,
> the
> > > > hypotenuse of a right triangle.

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:

> A mind reader I am not, however, and not a single post of
> the "Uncertainty Principle" thread (including the original post)
> clearly states the actual spectral situation to be modeled. I have
> asked Paul a couple of times in this thread to clarify the
situation
> which he envisions, but I guess he missed those requests for info.

Well, I thought I clarified in my posts to Bob. Since we're talking
about perceiving _different_ pitches, succeeding one another in time,
I suspect a meaningful modeling of the situation I'm interested in
would have to involve something like a wavelet transform (thanks Dave
Keenan) -- if you know much about these, please respond over at

tuning-math@yahoogroups.com

P.S. Gene -- I intend to take a crack at your puzzle next week, and
I'll do it at

tuning-math@yahoogroups.com
.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "J Gill" <JGill99@i...> wrote:
>
> > A mind reader I am not, however, and not a single post of
> > the "Uncertainty Principle" thread (including the original post)
> > clearly states the actual spectral situation to be modeled. I
have
> > asked Paul a couple of times in this thread to clarify the
> situation
> > which he envisions, but I guess he missed those requests for info.
>
> Well, I thought I clarified in my posts to Bob. Since we're talking
> about perceiving _different_ pitches, succeeding one another in
time,
> I suspect a meaningful modeling of the situation I'm interested in
> would have to involve something like a wavelet transform (thanks
Dave
> Keenan) -- if you know much about these, please respond over at
>
> tuning-math@y...

J Gill:

Paul, I don't know much about the Wavelet transforms to have much to
say (I read that one can "have one's cake and eat it, to, that is -
be able to accurately resolve small bandwidths_without_having to
average many cycles, as is necessary with bandpass filters, DFTs,
FFTs, and many other similar frequency domain transforms).

If you guys are talking about the case of successive (non-
simultaneously sounded) tones (sine waves or with harmonic multiples
present as well as the fundamental?) only, it seems like it might be
a stretch to speculate about a "tonal memory mechanism" which would
expect any particular pitch to occur next, would be anticipating a JI
pitch in lieu of a 72-tET pitch (would not that ascribe some kind of
preference for one or the other of these systems, itself?), and, with
the previous tone somehow mapped into "mind" (as if on a tape loop),
compare the memory of such a previous tone with the new tone in some
sort of spectral analysis (of two tones, one of which is not even
sounded at that time). How can these independent acoustical events be
meaningfully compared to each other?

Curiously, J Gill

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:
>
> If you guys are talking about the case of successive (non-
> simultaneously sounded) tones (sine waves or with harmonic multiples
> present as well as the fundamental?)

We're thinking of inharmonic timbres or sine waves, because harmonics of course would reduce
the "uncertainty" in proportion to the harmonic #.

> only, it seems like it might be
> a stretch to speculate about a "tonal memory mechanism" which would
> expect any particular pitch to occur next, would be anticipating a JI
> pitch in lieu of a 72-tET pitch

Hmm . . . that's not really within the scope of this discussion, but there are those around here
who buy into the whole Boomsliter & Creel theory that we do prefer JI intervals in melody,
should they wish to speak up at this point . . .

. . . but no, the issue hear is one of "hearing the difference" between such a musical passage
(say, a hexadic arpeggio, or a utonal hexadic chord) in 72-tET vs. in JI.

> (would not that ascribe some kind of
> preference for one or the other of these systems, itself?), and, with
> the previous tone somehow mapped into "mind" (as if on a tape loop),
> compare the memory of such a previous tone with the new tone in some
> sort of spectral analysis (of two tones, one of which is not even
> sounded at that time).

That would be the Boomsliter & Creel tack.

> How can these independent acoustical events be
> meaningfully compared to each other?

If you're asking how the Boomsliter & Creel phenomena could take place, you're asking the
wrong person -- I don't even believe that they do (their experiment was anything but a true
controlled double-blind experiment with adequate alternatives tested).

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:

> With a clear idea of what exactly is being talked about, I would be
> interested in commenting (regarding the "Uncertainty thread, and FM
> spectra of complex modulating spectrums), and hopefully assisting
in
> the process.
>
>
> Best Regards, J Gill

Thanks, for the offer! I obviously could use the help. My math is
quite weak now, since I graduated in the late 60s and have used it
little since, except for analytical geometry, trig and basic algebra.

I believe Paul is just asking what happens when, instead of simply
introducing some bandwidth to a stable frequency with amplitude
modulation in the form of attack and decay envelopes, you run rapidly
up a scale legato (no starts or stops of the signal) so that a
sequence of pitches is generated. So effectively you are modulating
the frequency with a stepped wave.

Paul seems to feel that the assymetry of this frequency shifting
might introduce classical uncertainty intrinsic to pitch perception
given a sufficiently rapid change in pitch. I guess we're both
wondering if and at what point this would happen and if it does
happen, is the intrinsic uncertainty greater than the limitations
imposed by human hearing or negligible before the limits of human
hearing are reached.

My argument was that the sideband components of such frequency
modulation would not be closey related enough to the pithes under
consideration to introduce classical uncertainty, and if the rapidity
of the stepped changes were high enough to generate separately
audible sideband components (which I believe would eventually
appear), they would be musically extraneous and the whole question
would become moot, since this would happen only at a rapidity that
exceeds the bounds of any non-electronic performance parameters.

Paul (post #29677)

"We're thinking of inharmonic timbres or sine waves,.."

"...the issue hear is one of "hearing the difference" between such a
musical passage (say, a hexadic arpeggio, or a utonal hexadic chord)
in 72-tET vs. in JI."

J Gill:
So, it is the case of an arpeggio (of sine wave), OR the case of a
chord with simultaneous sounding of (sine waves)?

Bob (post #29713):

"I believe Paul is just asking what happens when, instead of simply
introducing some bandwidth to a stable frequency with amplitude
modulation in the form of attack and decay envelopes, you run rapidly
up a scale legato (no starts or stops of the signal) so that a
sequence of pitches is generated. So effectively you are modulating
the frequency with a stepped wave.
Paul seems to feel that the assymetry of this frequency shifting..."

J Gill: Have found some relevant material in "Spectrum Analyzer
Measurements: Theory and Practice", Engelson, Tektronix, regarding
the non-linear world of FM by complex (non-sinusoidal) modulating
waveforms:

<<... the spectrum of multitone AM is just the sum of the individual
single-tone spectra. In multitone FM, there is an interaction between
the several modulating frequencies, creating additional sidebands
than is apparent by treating each tone individually. The mathematics
for multitone FM can get quite complicated and will not be reproduced
here" [footnote of Giacoletto, "Generalized Theory of Multitone AM
and FM", Proc IRE July 1947].

A major difference between the spectra of single-tone and multitone
FM is that while in the former the sideband distribution is
symmetrical around the carrier, in the latter it need not be. While
an absolute rule is difficult to formulate, because of the complexity
of the situation, it has generally been found that symmetrical
modulating waveshapes create symmetrical spectra while unsymmetrical
modulating waveshapes create unsymmetrical spectra.

In multitone, as in single-tone FM, the total energy is constant
regardless of the degree of modulation. Hence, in multitone FM, as
the number of sidebands is increased the carrier component is
decreased.>>

J Gill notes:
The text above is describing a rule of thumb applying to_periodic,
repeated_unsymmetrical modulating waveshapes creating unsymmetrical
frequency modulated spectra. The case of a number of discrete steps
approaching a (stepped) "ramp-like" increase (or decrease) in pitch,
in a manner analogous to the considerations of the Fourier integral
(for single transient events) as opposed to the Fourier transform
(for steady-state periodic spectra), carries with it the added
spectral complexity of the length as well as shape of it's amplitude
envelope (aka "time-domain window") of the - frequency modulating -
function of time, itself, which resultes in additional (sin f)/f
continuous spectral energy surrounding each of the individual
spectral components of the - frequency modulating - spectra, as a
result of the (effective) amplitude modulation of the - frequency
modulating - function. Simple, eh?...

It_may_be reasonable to speculate that: a (either increasing or
decreasing) "ramp" - frequency modulating - function, being an even
(non-symmetric around t=0) function of time, and (in the periodic
repeating case) consisting of even (as well as odd) harmonics (of the
fundamental frequency of the - frequency modulating - waveform) which
result in asymmetrical time domain waveforms, would (as a non-
symmetrical modulating_transient_waveform likely result in a non-
symmetrical result frequency-modulated (FM) spectrum.

Care must be taken, however, to resist the temptation to think of
such "linear-swept-frequency" (or geometric-swept-frequency, if you
will) in what would seem to be the "intuitive" speculation that the
resultant spectral energy actually exists for a time at each
small "step" along the way of such a "swept-frequency" carrier
(frequency modulated higher frequency) in the course of occurance of
such an "arpeggio"... Paradoxically, FM and PM confound our sense of
(spectral) intuition, best put by Engelson below:

<< We shall not go through the derivation of the network response
leading to the remarkable fact that the total output of a narrowband
filter with an FM input has energy only at discrete frequencies
[footnote Harvey, et al, "The Component Theory of Calculating Radio
Spectra with Special Reference to FM" Proc IRE June 1951]. While the
input [the - frequency modulating - function, J Gill] consists of a
time-variable signal going through all the frequencies between F - dF
and F + dF the output of a narrowband filter, consisting of the
combined transient and steady-state response [of the narrowband
filter through which a frequency modulated signal is passed, J Gill]
and averaged over one FM cycle of interval T [T=time period of one
cycle of a_repeated_modulating waveform, J Gill], contains no energy
except at the frequencies indicated by accepted FM theory.

...it is not our intention to resolve the question of whether the
spectral components are a part of the signal or are generated by the
circuit. The important thing to remember is that real, physically
realizable, linear, time-invariant circuits behave as if spectral
components exist, and this is what we wish the spectrum analyzer to
show.>>

J Gill:
This stuff really makes one wonder whether the spectral components of
even such a frequency modulated_sine wave_(ie geometric arpeggio) are
determinable by man or machine in a manner which readily comports
with our (necessarily) simplistic descriptions of a single, "average
pitch" model, when , in fact, the spectral "forest" is_dense_and
overlapping (as opposed to being made up of easily separable spectral
lines. It is interesting that the mathematics appear to imply the
existence of some kind of "narrowband" (hence, slow to respond to a
high level of accuracy) filter occuring_somewhere_in the "signal-
brain" chain (ie spectrum analyzer performing a Fourier integral) as
a requisite for the characterization of the relative energies of the
spectral components of the frequency-modulated (FM) sinusoidal wave
in question. Whether our sense of hearing is able to transcend such
limitations as they exist for other physical systems is debatable.
A 1 HZ bandwidth, as Paul tentatively appears to posit, may, indeed
(save for possible Wavelet magic) require Q/PI cycles to meaningfully
resolve (where Q = Fcenter/Bandwidth of a second order bandpass
element).

Noting that the average time for pitch recognition for individual
tones (Olson, "Music, Physics, and Engineering") remains roughly
constant at about 13 milliseconds (which constitutes about 3 or more
cycles of the pitch at the lower frequencies of hearing, and only
_increases_in "necessary cycles" as pitch is increased upward
throughout the audio bandwidth), it seems (perhaps) possible that
arpeggios of up to about 77 notes/second could (potentially) be
resolved by our "organic" apparatus. Yet, it is unclear to what
accuracy these "pitch recognitions" were executed in the generation
of the data referenced in Olson, above.

Bob (continued):
"...might introduce classical uncertainty intrinsic to pitch
perception
given a sufficiently rapid change in pitch."

J Gill: In addition to the spectral asymmetry issue discussed above,
Bob appears to have also introduced the spectral widening (increase
of the distribution of spectral FM sideband energy outward, above and
below the frequency modulated) which occurs as the "FM modulation
index" (equal to dFc/Fm, the ratio of the deviation of the carrier
frequency over the frequency of modulation) itself_increases_to
values greater than unity. I would like to point out here that an
increase in the rate of a "geometric arpeggio", which in turn
increases the rise-time of the "ramp" - frequency modulating -
function, appears to actually_decrease_the "FM modulation index"
(being a decrease of the denominator of the ratio).

Only an increase in the resultant_deviation_(vibrato, or geometric
scale factor) of the_modulated_pitch would appear to increase the "FM
modulation index" (being an increase in the denominator of the
ratio), thus spreading the constant energy of the FM modulated pitch
out over upper and lower (possibly unsymmetrically distributed)
sideband pairs determined by the Bessel functions which describe FM.

The case of chords consisting of simultaneously sounded sine waves
which Paul has (appeared) to indicate in his post #29677 seems to me
to be rather different, and, while not involving energy at harmonics
of the sinusoidal components of a multi-tone chord, still may
(perhaps)involve human "spectral expectations" (ie, of harmonics of
one of the chords sinusoidal components) of a certain pitch being
present, due to the simultaneous presence of other pitches). How
such "spectral expectations" (if they exist) would inter-relate with
the process of resolving a pitch to a given accuracy level seems
unclear, but possibly not without significance in the process.

Further, if "wavelet" transforms really are (and I believe they are)
able to resolve arbitrarily narrow bandwidths when processing a
_single cycle_of the spectral component to be analyzed (as opposed to
multiple cycles, as in Laplace transform variants Fourier integral,
DFT, FFT, Z transforms, and numerous other frequency domain
transforms developed prior to wavelet theory), we here have
an "inorganic" example of a process which may provide clues as to the
possibility of "organic" pitch perception mechanisms equalling their
impressive performance in practice. After all, nature has had a lot
longer to build prototypes than have late 20th century mathematicians!

Regards, J Gill

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:

<< I would like to point out here that an
increase in the rate of a "geometric arpeggio", which in turn
increases the rise-time of the "ramp" - frequency modulating -
function, appears to actually_decrease_the "FM modulation index"
(being a decrease of the denominator of the ratio).

Only an increase in the resultant_deviation_(vibrato, or geometric
scale factor) of the_modulated_pitch would appear to increase the "FM
modulation index" (being an increase in the denominator of the
ratio), thus spreading the constant energy of the FM modulated pitch
out over upper and lower (possibly unsymmetrically distributed)
sideband pairs determined by the Bessel functions which describe FM.>>

WHICH SHOULD CORRECTLY STATE (CHANGES CAPITALIZED):

I would like to point out here that an
increase in the rate of a "geometric arpeggio", which in turn
increases the rise-time of the "ramp" - frequency modulating -
function, appears to actually_decrease_the "FM modulation index"
(being AN INCREASE of the denominator of the ratio).

Only an increase in the resultant_deviation_(vibrato, or geometric
scale factor) of the_modulated_pitch would appear to increase the "FM
modulation index" (being an increase in the NUMERATOR of the
ratio), thus spreading the constant energy of the FM modulated pitch
out over upper and lower (possibly unsymmetrically distributed)
sideband pairs determined by the Bessel functions which describe FM.

Regards, J Gill

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:

While the
input [the - frequency modulating - function, J Gill] consists of a
time-variable signal going through all the frequencies between F - dF
and F + dF the output of a narrowband filter, consisting of the
combined transient and steady-state response [of the narrowband
filter through which a frequency modulated signal is passed, J Gill]
and averaged over one FM cycle of interval T [T=time period of one
cycle of a_repeated_modulating waveform, J Gill], contains no energy
except at the frequencies indicated by accepted FM theory.

WHICH SHOULD CORRECTLY STATE (CHANGES CAPITALIZED):

While the
input [the FREQUENCY MODULATED SINE-WAVE, J Gill] consists of a
time-variable signal going through all the frequencies between F - dF
and F + dF the output of a narrowband filter, consisting of the
combined transient and steady-state response [of the narrowband
filter through which a frequency modulated signal is passed, J Gill]
and averaged over one FM cycle of interval T [T=time period of one
cycle of a_repeated_modulating waveform, J Gill], contains no energy
except at the frequencies indicated by accepted FM theory.

Regards, J Gill

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:

Further, if "wavelet" transforms really are (and I believe they are)
able to resolve arbitrarily narrow bandwidths when processing a
_single cycle_of the spectral component to be analyzed (as opposed to
multiple cycles, as in Laplace transform variants Fourier integral,
DFT, FFT, Z transforms, and numerous other frequency domain
transforms developed prior to wavelet theory), we here have
an "inorganic" example of a process which may provide clues as to the
possibility of "organic" pitch perception mechanisms equalling their
impressive performance in practice. After all, nature has had a lot
longer to build prototypes than have late 20th century mathematicians!

J Gill:

I would like to make a point to state that "Laplace transform
variants DFT, FFT, Z transforms", themselves, require, in theory,
only one single cycle of the lowest frequency to be analyzed in order
to generate an output, as well. However, the resulting "resolution
bandwidth" is inadequate (for most purposes) unless several cycles of
the periodicity to be analyzed are collected and averaged, in a
manner similar to the rise-time delay of the amplitude envelope of
the output of a bandpass filter element.

It seems that (and perhaps Dave Keenan would correct me if I am wrong
here in my speculation) the "wavelet transforms" offer, in essence,
more "resolution bandwidth" possible, without requiring_multiple
cycles_of the frequency to be analyzed to be averaged in order to
realize such an improved "resolution bandwidth". Any wavelet experts
out there?...

Regards, J Gill

Regards, J Gill

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:
> Paul (post #29677)
>
> "We're thinking of inharmonic timbres or sine waves,.."
>
> "...the issue hear is one of "hearing the difference" between such a
> musical passage (say, a hexadic arpeggio, or a utonal hexadic
chord)
> in 72-tET vs. in JI."
>
> J Gill:
> So, it is the case of an arpeggio (of sine wave), OR the case of a
> chord with simultaneous sounding of (sine waves)?

Well I would exclude the otonal simultaneity, since other phenomena
come into play there, but I would definitely include the utonal
simultaneity, as long as there are no harmonic partials in the
timbres.

> The case of chords consisting of simultaneously sounded sine waves
> which Paul has (appeared) to indicate in his post #29677 seems to
me
> to be rather different, and, while not involving energy at
harmonics
> of the sinusoidal components of a multi-tone chord, still may
> (perhaps)involve human "spectral expectations" (ie, of harmonics of
> one of the chords sinusoidal components) of a certain pitch being
> present, due to the simultaneous presence of other pitches).

I doubt it, if I'm understanding you correctly.
>
> Further, if "wavelet" transforms really are (and I believe they
are)
> able to resolve arbitrarily narrow bandwidths when processing a
> _single cycle_of the spectral component to be analyzed

I don't think that's correct. I believe wavelet transforms obey the
very same Classical Uncertainty Principle. However, I believe they
consistute a better model than Fourier transforms for how we hear a
succession of pitches -- clearly, when alternating slowly between two
pitches, we don't hear the center pitch, we hear an alternation
between two pitches. The question is, how precise is our perception
of _each_ of these two pitches?

Hey guys, unless I missed something important,
you seem to have veered off-topic from this thread's
subject line. Does any of today's discussion under
this subject have anything to do with my original
question about quadratics and Sumero-Babylonian tuning?

-monz

----- Original Message -----
From: Paul Erlich <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Tuesday, October 30, 2001 12:12 PM
Subject: [tuning] Re: Question about Quadratics

> --- In tuning@y..., "J Gill" <JGill99@i...> wrote:
> > Paul (post #29677)
> >
> > "We're thinking of inharmonic timbres or sine waves,.."
> >
> > "...the issue hear is one of "hearing the difference" between such a
> > musical passage (say, a hexadic arpeggio, or a utonal hexadic
> chord)
> > in 72-tET vs. in JI."
> >
> > J Gill:
> > So, it is the case of an arpeggio (of sine wave), OR the case of a
> > chord with simultaneous sounding of (sine waves)?
>
> Well I would exclude the otonal simultaneity, since other phenomena
> come into play there, but I would definitely include the utonal
> simultaneity, as long as there are no harmonic partials in the
> timbres.
>
> > The case of chords consisting of simultaneously sounded sine waves
> > which Paul has (appeared) to indicate in his post #29677 seems to
> me
> > to be rather different, and, while not involving energy at
> harmonics
> > of the sinusoidal components of a multi-tone chord, still may
> > (perhaps)involve human "spectral expectations" (ie, of harmonics of
> > one of the chords sinusoidal components) of a certain pitch being
> > present, due to the simultaneous presence of other pitches).
>
> I doubt it, if I'm understanding you correctly.
> >
> > Further, if "wavelet" transforms really are (and I believe they
> are)
> > able to resolve arbitrarily narrow bandwidths when processing a
> > _single cycle_of the spectral component to be analyzed
>
> I don't think that's correct. I believe wavelet transforms obey the
> very same Classical Uncertainty Principle. However, I believe they
> consistute a better model than Fourier transforms for how we hear a
> succession of pitches -- clearly, when alternating slowly between two
> pitches, we don't hear the center pitch, we hear an alternation
> between two pitches. The question is, how precise is our perception
> of _each_ of these two pitches?
>
>
>
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--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> I believe wavelet transforms obey the
> very same Classical Uncertainty Principle. However, I believe they
> consistute a better model than Fourier transforms for how we hear a
> succession of pitches...

How so, Paul?

Curiously, J Gill

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > I believe wavelet transforms obey the
> > very same Classical Uncertainty Principle. However, I believe
they
> > consistute a better model than Fourier transforms for how we hear
a
> > succession of pitches...
>
> How so, Paul?
>
>
> Curiously, J Gill

A Fourier transform simply gives a "static" representation of the
frequencies present in a signal, since these frequencies are assumed
to be of infinite duration. A decomposition into wavelets, on the
other hand, allows the frequency components to change over time --
the elements of wavelet basis are functions of finite support on the
time-axis.

We hear frequencies as changing over time -- thus my statement. I
know virtually nothing else about wavelets since my Analysis prof
decided to skip the topic (I rarely attended class anyway).