The Classical Uncertainty Principle and Tuning

I'm posting this more as a springboard for debate than as an actual
belief on my part. But it seems like something someone could
successfully argue, so I wonder what counterarguments there might be.

The classical uncertainty principle is a mathematical fact (unlike
the quantum-mechanical Heisenberg uncertainty principle, which is an
unexpected feature of nature). It derives from Fourier theory, and
basically says that a musical note, played for time t, can have its
frequency determined to no better than 2*pi/t (see
http://sepwww.stanford.edu/sep/prof/fgdp/c4/paper_html/node2.html --
equation 1).

Much has been made of the 3- and 4- cent deviations from JI in 72-tET
chords. At a typical musical frequency of 440Hz, 4 cents is a 1Hz
deviation. So the classical uncertainty principle would seem to say
that, for frequency to be determined to better than this accuracy,
the note would have to be played for 2*pi, or over 6 seconds! Clearly
most music has melodies and even chord changes that are much faster
than this. Thus any attempt to say whether the chords were in JI or
in 72-tET would be meaningless.

Rebuttals?

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

> Rebuttals?

Most people don't have absolute pitch in any case, so the relevant
question is the relationship between two or more tones. Two tones
1 Hz apart at 440 and 441 Hz are certainly not going to sound the
same as one tone played by itself.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Rebuttals?
>
> Most people don't have absolute pitch in any case, so the relevant
> question is the relationship between two or more tones. Two tones
> 1 Hz apart at 440 and 441 Hz are certainly not going to sound the
> same as one tone played by itself.

It'll sound like a 440.5 Hz tone, beating very slowly. So I'm not
sure what that proves. You may be thinking of beating between
partials as the next logical point in this argument. But I'm
thinking, for example, about Alison tuning her metal tubes to either
72-tET or JI, and playing basic chords. There you won't get any
beating between partials that could help you rate the JI tuning as
better than the ET tuning . . . so, if you're playing each note for
shorter than 6 seconds, isn't it true that there's no meaningful
reason to prefer the JI tuning?

Paul,

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> so, if you're playing each note for
> shorter than 6 seconds, isn't it true that there's no meaningful
> reason to prefer the JI tuning?

Or to prefer the ET tuning. If it doesn't matter, then the choice is
up to the person doing the creating, and it could very well end up
being a philosophical decision, which no amount of 'principles' can
support or rebut.

Makes one wonder if making a philosophical decision is "meaningfull",
no?

Cheers,
Jon

Hi Paul,

Yes, FTS can do much better than this by counting waves.

If one counts zero crossing points, then one can interpolate
to find it to a fraction of a sample.

E.g. if one sample is + 10 and next one is -40, using linear
interpolation (which is quite reasonable for most waves at
the crossing points) you know that the zero crossing point is
0.2 samples after the first of the two samples.

So over say a 0.1 sec interval at 44100 samples per sec, suppose
one can find the length of a wave train to say, perhaps quarter of a sample, then one has
an error of 1/10000 of the freq. I.e. 1.0001 or a bit over 0.1 cents.

Most wave samples fluctuate over that range of pitch.
I have just retested this by measuring a reasonably steady wave &
I get it steady to within +- 0.025 cents with wave counts of trains
of 0.1 seconds.

Freq. could be confirmed by wave count of 30 secs, with an FFT
of 30 secs also falling within that range.

As you see, one can improve the accuracy of FFT too.
FTS does this with with peak interpolation,
and one can certainly do better than 4 cents for 6 second
clip in that way. I wonder if that figure is quite correct,
as it doesn't seem to fit in with actual experiments.

Here are some results of my FFT experiments in FFT which I posted to
crazy_music a while back:

"
When you use the standard FFT algorithm, it produces frequency
bins, and for short sound clips the bins are so far apart as to give
very low frequency resolution. For instance, an 0.1 second clip at 440 Hz
gives a freq resolution of +- 10 cents - not very good!

However, one can improve the accuracy considerably, maybe achieve a
tenfold improvement of accuracy, by using peak interpolation. The method is
to look not just at the highest amplitude of the frequency bins, but also
the two bins to either side. Then by fitting a curve to it, you can estimate
where the real peak should be, on assumption that it is a response to a
single frequency partial.

I find it gives quite reasonable accuracy even for fairly short notes, say,
a tenth of a second. One can test it by making a waveform with some random frequencies
accurately pitched (and between FFT frequeny bins), and then see
if the FFT with the peak interpolation recovers them, and FTS has this
programmed in as a test to try out to see how well the FFT works.

To give a rough idea - of the five interpolation methods I've tried, I find
Jain's method is the one that seems to work best for the frequencies one is
interested in.

For tenth second random frequencies at around 440 Hz, it often gets the
pitch accurate to within 0.1 cents, but occasionally goes wild by about
1 to 2 cents. So ten times better accuaracy consistently, and most of the time
it has a hundred fold improvement in the accuracy. Pretty good!

For a 1 second clip, one can get better results, typically reliable to within
0.1 cents nearly all the time with Jain's method.
"

As you see, I tested this peak interpolation using a random superposition of
sine waves, with the wave to be tested actually constructed within FTS.
One can try out this experiment for oneself in current FTS beta preview
(though this section is still rather rough in the layout and user interface).

I haven't yet done the wave count tests in the same way - they were done
with an external "sine wave" generator, played through the soundcard and
then recorded in FTS.

Robert

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> I'm posting this more as a springboard for debate than as an actual
> belief on my part. But it seems like something someone could
> successfully argue, so I wonder what counterarguments there might
be.
>
> The classical uncertainty principle is a mathematical fact (unlike
> the quantum-mechanical Heisenberg uncertainty principle, which is
an
> unexpected feature of nature). It derives from Fourier theory, and
> basically says that a musical note, played for time t, can have its
> frequency determined to no better than 2*pi/t (see
> http://sepwww.stanford.edu/sep/prof/fgdp/c4/paper_html/node2.html --

> equation 1).
>
> Much has been made of the 3- and 4- cent deviations from JI in 72-
tET
> chords. At a typical musical frequency of 440Hz, 4 cents is a 1Hz
> deviation. So the classical uncertainty principle would seem to say
> that, for frequency to be determined to better than this accuracy,
> the note would have to be played for 2*pi, or over 6 seconds!
Clearly
> most music has melodies and even chord changes that are much faster
> than this. Thus any attempt to say whether the chords were in JI or
> in 72-tET would be meaningless.
>
> Rebuttals?

Paul,

The gist of your proposed hypothesis above appears (to me) to imply
that:

(1) A listener's perception of the_difference_in the pitch of two (or
possibly more) simultaneously sounded sinusoidal (or possibly more
spectrally complicated) tones is analogous to the process whereby a
listener perceives the individual pitches of each of such
(simultaneously sounded) individual tones; and, further that

(2) The process of a listener's perception is limited by the same
2*PI cycle length sampling window length which a 2nd order bandpass
filter network (and more sophisticated approaches utilizing Fourier
Integrals, FFTs, and Z Transforms) require in order to provide
results which can be considered reasonably valid.

Regarding (2) above:

Robert Walkers reply appears to explore some potentially interesting
possibilities for increasing the efficiency of certain "inorganic"
processes in assessing pitch (or perhaps pitches) more rapidly than
that of a time period of 2*PI cycles of the frequency.

Some information which may be relevant to "organic" pitch perception
mechanisms follows. Harry F Olson's "Music, Physics, and
Engineering", Dover, 1967, page 250, figure 7.6 depicts some data
regarding the "duration of a tone in cycles required to ascribe a
certain pitch (After Burck, Kotowski, and Lichte; von Bekesy and
Turnbull)". From the text which describes figure 7.6:

"The number of cycles required to ascribe a definite pitch to a tone
depends on the frequency, as shown in Fig. 7.6. It will be seen that
the number of cycles required to establish the pitch of a tone
increases with the frequency. At 50 cycles it is 3 cycles,.."[LESS
THAN 2*PI CYCLES] "at 400 cycles it is 7 cycles,.." [APPROX 2*PI
CYCLES] "at 1,000 cycles it is 12 cycles, and at 8,000 cycles it is
145 cycles. It may be noted that the time required to ascribe a
definite pitch does not vary a great deal from low to high
frequencies. The average time is about 13 milliseconds."

Figure 7.6 shows a roughly monotonic slope (in the_logarithm_of
duration in cycles) between 50 CPS and 400 CPS. It is interesting
that these findings imply that more may be going on in "organic"
pitch perception processes than our best machinery may be able to, in
this respect,(reliably) resolve...

Regarding (1) above:

Can it really be said that a listener does not perceive the pitch of
a 440 CPS tone in, if you would, (2*PI)/440 seconds, or 14.3 mSEC,
while perceiving the pitch of a second (simultaneously sounded)441
CPS tone in, if you would, (2*PI)/441 seconds, or 14.25 mSEC?

Can it really be said that any effects of the amplitude envelope (in
sound pressure level) varying at 1 CPS (due to any non-linear
phenomena involved in the process which may generate difference
frequencies) would not present themselves fully within 1/2 of one
cycle of the 1 CPS difference frequency, or within 500 mSEC?

It seems to me that the time required for a listener to
resolve_differences_between frequencies may well be a different
matter than the time required to resolve the individual frequencies
(without broaching the potentially more complicated effects of such
tones having harmonic as well as inharmonic spectral content).
An "inorganic" example of this might be the small (but measurable)
differential signal which exists between the amplitude envelopes
(measured by instantaneous peak detection) of the outputs of two
identical 2nd order bandpass filter networks both tuned to 440 CPS,
where one filter is excited by a gated 440 CPS sinusoidal waveform,
and the second filter is excited by a gated 441 CPS sinusoidal
waveform. Such a differential signal would appear within one half-
cycle of these_excitation_frequencies (1.135 mSEC), as opposed to
having to wait 2*PI cycles (6.283 SEC) of the_difference_frequency,
(which is a factor of 5535 times faster than the proposed delay).

Regards, J Gill

PS - Your "Forms of Tonality" paper is a thing of beauty. I really
like form and quality of your graphics, and your text has, as you had
predicted, been helpful to me in taking in some of the ideas with
which you have put forth. Thanks again for sending it along.

PSS - Thanks for posting the link to Ernst Terhardt's web site, at:
http://www.mmk.e-technik.tu-muenchen.de/persons/ter.html
which has some_really_interesting stuff.

My paragraph (2) in my post is more accurately stated as:

"(2) The process of a listener's perception is limited by the same
(2*(damping factor))/PI cycle length (amplitude envelope) rise-time
which 2nd order bandpass filter networks require (and higher order
bandpass filter networks of equal bandwidth do not deviate greatly
from)in order to provide a resultant output signal which can be
considered reasonably valid as an indicator of spectral energy
existing at the center frequency of that bandpass network, in a
manner similar to the requirements that the length of the time-window
within which a Fourier integral is performed (or the time samples to
be input into a FFT algorithm are gathered) be sufficiently long so
as to provide an adequate spectral frequency selectivity of the
frequency-domain transformed output information.".

Without meaning to oversimplify the process of hearing, or to trammel
upon existing findings regarding that process of which I may well be
unaware, it is my guess that (with or without difference frequencies
created by nonlinear processes in the chain) what may be going on in
the process of differentiating between a 440 CPS sine wave and a 441
CPS sine wave is not analogous to the (clearly much slower) task of
definitively resolving a 1 Cycle bandwidth (with an analog filter
system, or a frequency domain transform algorithm), but_may_be closer
to the output of a (perhaps) linear phase detector capable of
detecting (albeit small) phase differences betwen two signal
frequencies to be compared, which appear within 1/2 (as opposed to
2*PI) cycles of the individual. In a manner similar to a Phase Locked
Loop (PLL) servo system, phase difference information from a phase
detector tending to indicate a difference in the two signal
frequencies compared becomes available with a time delay much shorter
than the significantly longer time within which such a PLL, acting as
kind of nonlinear tracking bandpass filter (with a bandwidth which is
determined by the loop parameters) processes a sufficient number of
cycles of the signal frequencies compared to reliably accomplish a
(virtual) "locked" state, thus providing an output similar to a
narrow-band band pass filter.

Disclaimers:

I am not attempting here to advance a theory of hearing, or claim to
know much about what happens in our "heads" when we compare pitches.
I would be very interested in any research findings supporting the
hypothesis which Paul has presented, relating to an inverse
relationship measured between difference frequency and time of
recognition of such differences by research subjects.

This idea does_not_relate to binaural situations (where below about
800 CPS, I think, human ears are able to process left/right phase
differences, thus placing the source).

The PLL phase detector model described above is presented only as an
instance of a (inorganic)process where a comparison of the difference
in frequency of signals _isolatable_to parallel pathways can and does
output potentially useful (albeit noisy) information regarding the
existence of such differences at a time well before such a system can
be said to indicate with "certainty" that spectral energy exists
precisely within its (usually relatively) narrow bandwidth, in its
function as frequency detector. It may well be arguable whether
simultaneously sounded frequencies (in one ear only) actually are, in
fact, "isolatable" in a model of perception. I will leave that up to
those who possess a deeper understanding of such matters than I... :)

J Gill

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> >
> > > Rebuttals?
> >
> > Most people don't have absolute pitch in any case, so the
relevant
> > question is the relationship between two or more tones. Two tones
> > 1 Hz apart at 440 and 441 Hz are certainly not going to sound the
> > same as one tone played by itself.
>
> It'll sound like a 440.5 Hz tone, beating very slowly. So I'm not
> sure what that proves. You may be thinking of beating between
> partials as the next logical point in this argument. But I'm
> thinking, for example, about Alison tuning her metal tubes to
either
> 72-tET or JI, and playing basic chords. There you won't get any
> beating between partials that could help you rate the JI tuning as
> better than the ET tuning . . . so, if you're playing each note for
> shorter than 6 seconds, isn't it true that there's no meaningful
> reason to prefer the JI tuning?

Paul,

Are you saying directly (quoted) above that - after 6 seconds, in the
situation described, the differences between the original choices of
72-tET or JI may_become_apparent? How would the listener know what JI
frequency they had_expected_to hear (in order to require that at
least 2*PI cycles of the difference frequency between the JI pitch
and the 72-tET pitch transpire in order to notice the difference)?

Respectfully, J Gill

Hi Paul,

FFT bin size in Hz = 1/sample length in secs.

(often given as sample frequency/fft length in samples)

So for 1 sec clip, bin size is 1 Hz, which at 440 Hz, is a little under 4 cents.

(Note, this is for 1 sec clip. For 6 sec clip, then 4 cents corresponds to
about 6 bins).

However, that is without peak interpolation, just going for the
highest peak.

If one looks at the bins to either size, and one uses those to find
the position of the highest peak, one can do much better.

This can only improve accuracy by assuming that the FFT is locally
the result of a single frequency. However, usually that is the
exact situation one is interested in in a musical context.

E.g. if the greatest magnitude in the FFT is for the bin at say 440 Hz,
but the one at 441 Hz is greater than the one at 439 Hz, then the frequency
is a little over 440 hz. If the one at 439 Hz is the greater, then the
frequency is a little under 440 Hz.

|
| |
| | |
..439..440..441...

Freq is a little over 440 hz

|
| |
| | |
..439..440..441...

Freq is a little under 440 hz

Then, one knows well how the FFT responds to a single frequency, so using
this one can find the exact frequency with pretty good accuracy.

This has all been worked out and we have the likes of Jain's method
and Quinn's estimators to use to find the exact frequencies.

I find Jain's method is the one that works best for 440 hz range
frequencies, and it gives an accuracy of 0.1 cents for 1 sec.
clip.

In fact, if one tries it even for 0.1 sec clips, it gets it accurate
to within 0.1 cents most of the time. Only the occasional measurement
of a 0.1 sec clip goes wild at 2 or 3 cents out.

So on hypothesis that our hearing is modelled will via FFT, which is just a hypothesis,
then accuracy depends on how good the ear is at peak interpolation.

Perhaps those who are particularly good at hearing frequencies of notes have
learnt to do peak interpolation to find the frequencies?

Just one idea.

If one uses the wave crossing method, there is no limit on resolution at all
- the more samples per second, the better the time resolution, and the frequency
resolution gets better accordingly.

There's an interesting connection between QM. and FFT, as I'm told
that both dep. on the Sturm Louiville theorem. I'd like to learn
more about this.

However, there's also a difference. In QM, if one measures the
momentum of a particle, the data is lost as a result of the measurement,
and having chosen to collapse the wave to find the position of the
particle, one can't go back and find the momentum, or vice versa.
One has no access to the waveform data, just to the results of the
measurements, so having made the measurement, with choice to find
the position, one can't go back to find the momentum, or vice versa.

In FFT the waveform data is there, one has direct access to it,
and one can ananlyse and re-analyse it as often as one likes and in
as many ways as one likes.

I think it is true that if one listens to a longer note one can
hear the pitch more accurately. For example on listening to
two notes one after another of one second duration, I can more
easily hear which is the higher in pitch (if very close in pitch)
than if they are both of 0.1 secs duration.

Robert

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, October 17, 2001 4:32 PM
> Subject: [tuning] Re: The Classical Uncertainty Principle and Tuning
>
>
> --- In tuning@y..., genewardsmith@j... wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> >
> > > Rebuttals?
> >
> > Most people don't have absolute pitch in any case, so the relevant
> > question is the relationship between two or more tones. Two tones
> > 1 Hz apart at 440 and 441 Hz are certainly not going to sound the
> > same as one tone played by itself.
>
> It'll sound like a 440.5 Hz tone, beating very slowly. So I'm not
> sure what that proves. You may be thinking of beating between
> partials as the next logical point in this argument. But I'm
> thinking, for example, about Alison tuning her metal tubes to either
> 72-tET or JI, and playing basic chords. There you won't get any
> beating between partials that could help you rate the JI tuning as
> better than the ET tuning . . . so, if you're playing each note for
> shorter than 6 seconds, isn't it true that there's no meaningful
> reason to prefer the JI tuning?
>

Hi Paul and Gene,

Here's an interesting digression I'd like to note.

Upon reading this, I immediately thought of one
specific musical context -- the music composed and
performed by Kraig Grady -- where I think there would
be an audible difference between a JI and a 72-tET
tuning. Kraig's style sets up patterns of beating
between notes, which occurs over fairly long stretches
of time in the music, and which, I'm sure, would be
perceptibly different in those two tunings.

Comments from other Grady fans? Or Kraig himself?

love / peace / harmony ...

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> Paul,
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > so, if you're playing each note for
> > shorter than 6 seconds, isn't it true that there's no meaningful
> > reason to prefer the JI tuning?
>
> Or to prefer the ET tuning.

Sorry, Jon, it doesn't work both ways. For the JI tuning will
have 'wolves' that are larger than 4 cents.

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul,
>
> Yes, FTS can do much better than this by counting waves.

I know that, but is that really a meaningful measure of pitch?
Consider the response of the cochlear membrane to such a short wave-
pulse, for example.

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

> Paul,
>
> The gist of your proposed hypothesis above appears (to me) to imply
> that:
>
> (1) A listener's perception of the_difference_in the pitch of two
(or
> possibly more) simultaneously sounded sinusoidal (or possibly more
> spectrally complicated) tones is analogous to the process whereby a
> listener perceives the individual pitches of each of such
> (simultaneously sounded) individual tones;

No, I don't think the gist of what I was saying had anything to do
with simultaneous tones at all, especially not two tones at 440Hz and
441Hz, which would never both occur in the tuning systems in question.
>
> PS - Your "Forms of Tonality" paper is a thing of beauty. I really
> like form and quality of your graphics, and your text has, as you
had
> predicted, been helpful to me in taking in some of the ideas with
> which you have put forth. Thanks again for sending it along.

You're very welcome!
>
> PSS - Thanks for posting the link to Ernst Terhardt's web site, at:
> http://www.mmk.e-technik.tu-muenchen.de/persons/ter.html
> which has some_really_interesting stuff.

You bet!

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

> Paul,
>
> Are you saying directly (quoted) above that - after 6 seconds, in
the
> situation described, the differences between the original choices
of
> 72-tET or JI may_become_apparent? How would the listener know what
JI
> frequency they had_expected_to hear (in order to require that at
> least 2*PI cycles of the difference frequency between the JI pitch
> and the 72-tET pitch transpire in order to notice the difference)?

One example that comes to mind is Daniel Wolf, who had trained
himself to be able to recognize, and tune by ear, just ratios with
sine waves (say using two analog synthesizers controlled by dials).
He tended to take a very long time doing this . . .

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

> Upon reading this, I immediately thought of one
> specific musical context -- the music composed and
> performed by Kraig Grady -- where I think there would
> be an audible difference between a JI and a 72-tET
> tuning. Kraig's style sets up patterns of beating
> between notes, which occurs over fairly long stretches
> of time in the music, and which, I'm sure, would be
> perceptibly different in those two tunings.

Yes, the rates of beating would be slightly different. But this is
not the context in which the argument was put forward.

However, I'd be happy to apply the argument to claim that, if Kraig
were to play an 11-limit Utonal chord, one (even Daniel Wolf) could
not tell whether it was in JI or in 72-tET, unless the chord were
held for a long time (on the order of 2*pi seconds).

Paul,

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > Or to prefer the ET tuning.
>
> Sorry, Jon, it doesn't work both ways. For the JI tuning will
> have 'wolves' that are larger than 4 cents.

A. I am unclear on the above statement; maybe you can clarify the
point.

B. You didn't respond to anything else, regarding the philosophical
choices a composer/performer can make. Put down the oscilliscope
(sp?) and look elsewhere.

(Pt. 2 coming to another msg...)

Best,
Jon

Paul,

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> However, I'd be happy to apply the argument to claim that, if Kraig
> were to play an 11-limit Utonal chord, one (even Daniel Wolf) could
> not tell whether it was in JI or in 72-tET, unless the chord were
> held for a long time (on the order of 2*pi seconds).

Who cares??? If Daniel, or anyone else for that matter, couldn't tell
the difference, who cares whether it is JI or 72?

Once again, the realm of musical choice, taste, and philosophy is
asked to take a back seat to 'arguements'. I resist - constantly - to
respond to these kind of 'debates' (which aren't really debates, but
one person trying to find more 'proof' of one side being superior
over another), because I'll end up sounding like someone I don't like
in the least. But you can write all you want about why someone who
did something in JI could have done it -- what: better? cheaper?
lower fat content? -- in 72tET, and it will ALWAYS miss the point:
that was their choice, and there is *nothing* wrong with it.

I got better things to do, like make music.

Disgruntled, but soon to be gruntled again,
Jon

Hi Paul,

> > Yes, FTS can do much better than this by counting waves.

> I know that, but is that really a meaningful measure of pitch?

Yes, I say that it is. The wave count method accords with FFT
but is as accurate as an FFT with a longer sample.

- with a 0.1 sec wave count you get same result as about a 1 sec.
FFT with peak interpolation, or about 30 secs without it.

What about the rest of my post where I pointed out that you can
use FFT peak interpolation to get resolution of 0.1 cents for a
1 sec. clip?

Here are my ascii diagrams to show how peak interpolation works again.

|
| |
| | |
..439..440..441...

Freq is a little over 440 hz

|
| |
| | |
..439..440..441...

Freq is a little under 440 hz

This is actual practice, an established technique used in FFT.

I suggested, though just as an idea to consider, if FFT is indeed a good
model of how we perceive pitch, then this could be how those who have
better pitch resolution than 4 cents are able to distinguish such pitches.

Also encouraging for everyone if that is it since it is a matter
of learning, and practice, rather than some kind of innate ability.

There have been posts to this list by musicians who say they
can distinguish between notes just one cent apart or less.

If one can't explain something, ones model needs to be updated,
rather than hard to explain observations discarded.

This is just an example to show that one can have a model, for those
of scientific bent.

From previous discussions, I'm not sure if anyone knows exactly
how we percieve pitch anyway...

Seems a bit unlikely that we do a FFT algorithm on a stream of binary
data, so it is a very approximate model at that, if there is anything
to it at all.

Robert

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> Paul,
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > > Or to prefer the ET tuning.
> >
> > Sorry, Jon, it doesn't work both ways. For the JI tuning will
> > have 'wolves' that are larger than 4 cents.
>
> A. I am unclear on the above statement; maybe you can clarify the
> point.

The point is that in Blackjack (which was the context of this
discussion), or even in the diatonic scale, using JI ratios for all
the notes, in order to bring some of the chords into perfect just
tuning, will cause some of the chords to be out-of-tune, with errors
much worse than that in the original temperament proposed.
>
> B. You didn't respond to anything else, regarding the philosophical
> choices a composer/performer can make. Put down the oscilliscope
> (sp?) and look elsewhere.

I'm sorry I can't follow all the tangents this discussion has taken.
If you want to strike up a separate discussion about philosophical
choices a composer/performer can make, I'd be happy to do so. You may
also note that my original message in this thread (with a slightly
different subject line) began with the disclaimer: "This is not
something I actually believe".

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> Paul,
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > However, I'd be happy to apply the argument to claim that, if
Kraig
> > were to play an 11-limit Utonal chord, one (even Daniel Wolf)
could
> > not tell whether it was in JI or in 72-tET, unless the chord were
> > held for a long time (on the order of 2*pi seconds).
>
> Who cares??? If Daniel, or anyone else for that matter, couldn't
tell
> the difference, who cares whether it is JI or 72?

Because, for a given scale, 72 will allow you a greater number of
these consonant chords than JI.

> Once again, the realm of musical choice, taste, and philosophy is
> asked to take a back seat to 'arguements'.

What are you talking about? You asked them to take a back seat?

> I resist - constantly - to
> respond to these kind of 'debates' (which aren't really debates,
but
> one person trying to find more 'proof' of one side being superior
> over another),

Nonsense.

> because I'll end up sounding like someone I don't like
> in the least. But you can write all you want about why someone who
> did something in JI could have done it -- what: better? cheaper?
> lower fat content? -- in 72tET, and it will ALWAYS miss the point:
> that was their choice, and there is *nothing* wrong with it.

Geez, when did I EVER suggest that a musician who did something in JI
should have done it in 72tET instead?

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul,
>
> > > Yes, FTS can do much better than this by counting waves.
>
> > I know that, but is that really a meaningful measure of pitch?
>
> Yes, I say that it is. The wave count method accords with FFT
> but is as accurate as an FFT with a longer sample.

Right -- but in terms of HUMAN PERCEPTION of an ACTUAL MUSICAL
INSTRUMENT TONE lasting a fraction of a second -- that's what I meant.
>
> I suggested, though just as an idea to consider, if FFT is indeed a
good
> model of how we perceive pitch, then this could be how those who
have
> better pitch resolution than 4 cents are able to distinguish such
pitches.

I'll let Gene or someone else respond to this (because I don't think
it's correct though I'm a little rusty on the mathematics/physics),
but I found Jay Gill's rebuttal far more relevant and effective.

> There have been posts to this list by musicians who say they
> can distinguish between notes just one cent apart or less.

In certain cases with simultaneous tones, anyone can.

> If one can't explain something, ones model needs to be updated,
> rather than hard to explain observations discarded.

True . . . what I had in mind in the way of rebuttals were actual
psychoacoustical data and phenomena . . . such as those Jay
presented. It appeared to me initially that for some set of
circumstances where certain psychoacoustical phenomena don't have a
chance to play a role, the argument I presented might be relevant. By
no means did I propose to discard hard-to-explain observations!
>
> This is just an example to show that one can have a model, for those
> of scientific bent.
>
> From previous discussions, I'm not sure if anyone knows exactly
> how we percieve pitch anyway...
>
> Seems a bit unlikely that we do a FFT algorithm on a stream of
binary
> data, so it is a very approximate model at that, if there is
anything
> to it at all.

Robert, if you're interested in how we perceive pitch, I strongly
suggest you carefully read over Ernst Terhardt's webpages:
http://www.mmk.ei.tum.de/persons/ter.html

Paul,

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Because, for a given scale, 72 will allow you a greater number of
> these consonant chords than JI.

Maybe every composer doesn't need a greater number of consonant
chords; your implication is that fewer consonant chords is somehow
inferior, and that may or may not be the case. Much good music, as
I'm sure you would agree, has been made with a small pallette.

> Geez, when did I EVER suggest that a musician who did something in
> JI should have done it in 72tET instead?

I would say the implication is there in your statements: if the
differences in audibility is "meaningless", and (as you state above)
there is some resource in 72tET in abundance over some other system
such that you bring up the fact... why else?

I'm perfectly willing to say I (once again) may have completely been
flummoxed by your reasons for pursuing this line of ideas. It's been
known to happen.

Jon

Hi Paul,

The FFT results work; one can find pitches by FFT to within
0.1 cents for a one second clip. Better than that most of the
time in fact.

The explanation I gave, with the ascii diagram,
describes a standard technique.

Others are welcome to comment on it and to say anything that
I may have missed out or not got quite correct - if I have made
mistakes - I'd like to know. If so, it will be an error in
my explanation, but the result stands whatever.

Maybe it will help to use analogy of a CCD image. If a star
image spans two or three pixels, then one can locate the
exact point within that group of pixels.

E.g.

x x
x x

The star is at mid point of the four xs, i.e. between the
CCD camera pixels.

x X

the star is between the two pixels, and somewhat more towards the
brighter X.

It works a little like that. Not intended as an exact parallel,
just an explanation for any who find the graphs hard to interpret.

I can't say anything about how we actually do it, just replying
to your suggested argument by logic that any perception of less
than 4 cents of separate notes of one second duration one after
another is impossible.

To show that the argument doesn't work, one just needs to show
that the pitch can be found using FFT by some method or another.
This I have done, and do regularly as a practical thing.

It is possible to find the pitch from a 440 hz note of 1 second to
within 0.1 cents, and most of the time the same technique gets a
0.1 sec. note correct to 0.1 cents too.

Since the argument assumed nothing about pitch perception except that
it is based on fourier analysis in some form, then this shows that
the argument doesn't work, which is what you wanted to know.

This shows that the uncertainty principle can't be used to set a limit of 4 cents
on perception of notes of 1 second, and that is all I wanted to show.

I'm not going to try to give a physical model of how
it is actually done - not my field or something I have any plans
to devote what would surely be years of work to do, if it
can be done at all at the current stage of knowledge. Other things
to do that I'm better able and qualified and interested to do.

What one can do is to actually try it out and see if one can
distinguish two notes 4 cents apart each of one second.

Indeed, apparently some musicians can do the same
with notes only 1 cent apart.

I think this matter of virtual pitch perception is rather
beside the point. Here we are interested in very
simple type of perception. E.g. perception of pitch
of a sine wave would be fine.

I find recorder timbre is quite an easy one to use
for distinguishing pitches on the soundcard, and it is
pretty close to sine wave.

I get the impression nobody quite knows how we are able to
perceive the pitch of, say, a sine wave.

Hope you get this sorted out to your satisfaction.

Hope also no-one gets discouraged from developing their
pitch sensitivity further by this argument as it isn't
a valid one.

Robert

Well, I've been following this one with a certain keenness of
interest, as some of you more familiar with my opinions might
imagine. I'm known locally as something of an intonational
perfectionist to a fault, yet I have a lot of trouble absorbing the
idea that 4 and 5 cent pitch errors on eighth or sixteenth notes at
moderate tempos are detectable by mortal ears, Johnny Reinhard
notwithstanding.

I don't want to be dogmatic and deny out of hand any possibility that
Johnny may be accurate in his assessments, but I do have trouble
absorbing them, frankly, since they fly in the face of my personal
experience and that of the finest ears whose functioning I have ever
had the privilege of witnessing, including my father's, whose hears
could set the temperament and tune a piano in an hour to a standard
equaled by few professional ET tuners and have it hold for amazingly
long times.

I do not advance my opionion on the basis of any mathematical
principles, although I welcome their rigor in substantiating
practical experience. My opinion here is strictly based on my own
practical experience, having spent hours over many years tweeking
just intervals to the nth degree for my own aural enlightenment and
education, and having started with what could only be assessed as an
already unusually accurate set of ears even among professionals.

I, like most, I believe, can more easily detect an error of 5 cents
when demonstrated as a harmonic interval, than as a melodic one in
which the two tones are sounded sequentially. Nonetheless, a meantone
perfect fifth, for example (5.38 cents flat), does sound flat to me
even MELODICALLY as long as the two tones are sounded in a context
that introduces a low level of cognitive interference with the
perception.

This means that if both tones are sounded, one immediately following
the other and there is, say, a half note on the second tone or maybe
slightly less at an andante tempo before a third tone is sounded, I
hear the flatness. In a rapid scale or arpeggio do I hear it? If the
arpeggio is sustained, yes; otherwise, frankly no, I don't. But on
the other hand I have NO trouble distinguishing such a run in JI or
meantone from one in 12-tET. The difference is immediately obvious
and NOT AT ALL SUBTLE.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "J Gill" <JGill99@i...> wrote:
>
> > Paul,
> >
> > Are you saying directly (quoted) above that - after 6 seconds, in
> the
> > situation described, the differences between the original choices
> of
> > 72-tET or JI may_become_apparent? How would the listener know
what
> JI
> > frequency they had_expected_to hear (in order to require that at
> > least 2*PI cycles of the difference frequency between the JI
pitch
> > and the 72-tET pitch transpire in order to notice the difference)?
>
> One example that comes to mind is Daniel Wolf, who had trained
> himself to be able to recognize, and tune by ear, just ratios with
> sine waves (say using two analog synthesizers controlled by dials).
> He tended to take a very long time doing this . . .

Jon, is it not perfectly legitimate and even important that we spell
out the differences and relative properties, leaving the choice up to
the creative desires of the musician/composer? How does doing this
automatically imply all the things you say Paul has implied?

Shouldn't we look for some basis for our choices rather than just
blind, random chance? I'll either leave chance music to John Cage or
consciously choose to make it chance music rather than be forced to
make it so out of ignorance of the consequences of my choices.

In brief, I don't think there is any need for any of us to be quite
so defensive of our right to choose. I can see nothing in simply
comparing properties that would challenge that right. If you like
dissonance, then use it, and if you're going to do that, it's better
to know it's there, isn't it?

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> Paul,
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > Because, for a given scale, 72 will allow you a greater number of
> > these consonant chords than JI.
>
> Maybe every composer doesn't need a greater number of consonant
> chords; your implication is that fewer consonant chords is somehow
> inferior, and that may or may not be the case. Much good music, as
> I'm sure you would agree, has been made with a small pallette.
>
> > Geez, when did I EVER suggest that a musician who did something
in
> > JI should have done it in 72tET instead?
>
> I would say the implication is there in your statements: if the
> differences in audibility is "meaningless", and (as you state
above)
> there is some resource in 72tET in abundance over some other system
> such that you bring up the fact... why else?
>
> I'm perfectly willing to say I (once again) may have completely
been
> flummoxed by your reasons for pursuing this line of ideas. It's
been
> known to happen.
>
> Jon

Interesting, Robert! Thank you. In a recent post to this thread, I
stated that for short durations in a scale passage, say in rapidly
running thirds, a difference of even 5 cents is difficult for me
personally to detect. Of course in this case, we're dealing with a
lot of cognitive interference, since other "trials" are occurring
both immediately before and after each harmonic interval in rapid
succession.

This only shows limitations in the perceptual apparatus and not one
of mathematical principle. In the same post, I clearly implied that I
could easily distinguish a 5 cent error even melodically in much less
than a second, so that already takes us beyond what this mathematical
argument would allow, considering the shortness of the time frame.

In this context, though, I must say I have heard music samples
referred to here and there in this list claiming to use tuning
systems the intonational accuracies of which were grossly violated by
singers and/or instrumentalists involved in the performance. It seems
awfully useless to me to get into such nit-picking discussions (and I
do dearly love them) if we're going to do them such violence in
practice. (Please don't ask me to be specific. I don't want to get
into that level of personal interaction on this subject.)

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul,
>
> The FFT results work; one can find pitches by FFT to within
> 0.1 cents for a one second clip. Better than that most of the
> time in fact.
>
> The explanation I gave, with the ascii diagram,
> describes a standard technique.
>
> Others are welcome to comment on it and to say anything that
> I may have missed out or not got quite correct - if I have made
> mistakes - I'd like to know. If so, it will be an error in
> my explanation, but the result stands whatever.
>
> Maybe it will help to use analogy of a CCD image. If a star
> image spans two or three pixels, then one can locate the
> exact point within that group of pixels.
>
> E.g.
>
> x x
> x x
>
> The star is at mid point of the four xs, i.e. between the
> CCD camera pixels.
>
>
> x X
>
> the star is between the two pixels, and somewhat more towards the
> brighter X.
>
> It works a little like that. Not intended as an exact parallel,
> just an explanation for any who find the graphs hard to interpret.
>
> I can't say anything about how we actually do it, just replying
> to your suggested argument by logic that any perception of less
> than 4 cents of separate notes of one second duration one after
> another is impossible.
>
> To show that the argument doesn't work, one just needs to show
> that the pitch can be found using FFT by some method or another.
> This I have done, and do regularly as a practical thing.
>
> It is possible to find the pitch from a 440 hz note of 1 second to
> within 0.1 cents, and most of the time the same technique gets a
> 0.1 sec. note correct to 0.1 cents too.
>
> Since the argument assumed nothing about pitch perception except
that
> it is based on fourier analysis in some form, then this shows that
> the argument doesn't work, which is what you wanted to know.
>
> This shows that the uncertainty principle can't be used to set a
limit of 4 cents
> on perception of notes of 1 second, and that is all I wanted to
show.
>
> I'm not going to try to give a physical model of how
> it is actually done - not my field or something I have any plans
> to devote what would surely be years of work to do, if it
> can be done at all at the current stage of knowledge. Other things
> to do that I'm better able and qualified and interested to do.
>
> What one can do is to actually try it out and see if one can
> distinguish two notes 4 cents apart each of one second.
>
> Indeed, apparently some musicians can do the same
> with notes only 1 cent apart.
>
> I think this matter of virtual pitch perception is rather
> beside the point. Here we are interested in very
> simple type of perception. E.g. perception of pitch
> of a sine wave would be fine.
>
> I find recorder timbre is quite an easy one to use
> for distinguishing pitches on the soundcard, and it is
> pretty close to sine wave.
>
> I get the impression nobody quite knows how we are able to
> perceive the pitch of, say, a sine wave.
>
> Hope you get this sorted out to your satisfaction.
>
> Hope also no-one gets discouraged from developing their
> pitch sensitivity further by this argument as it isn't
> a valid one.
>
> Robert

I just went back again to the paper from which this discussion took
its basis and keyed in on something I'd overlooked. The whole thesis
of frequency uncertainty is based on the SPECTRAL BANDWIDTH of tones
that are amplitude modulated in ways that can be described by attack
and decay envelopes plotted against time. Clearly, many if not most
musical tones, especially short ones that are percussively or
"pluckingly" produced, have significant frequency bandwidth
broadening due to attack and decay envelopes.

The argument that Paul has cleverly posited is based on the
assumption that spectral bandwidth is synonymous with uncertainty
concerning the frequency modulated by the attack and decay envelopes
in question. This assumption is fundmentally flawed. Bandwidth very
frequently has NOTHING to do with uncertainty as to the frequency
being modulated to produce that bandwidth. The bandwidth introduced
by such amplitude modulations is INHERENTLY SYMMETRICAL around the
base frequency. Psychoacoustically the center frequency is still
quite perceptibly the same as it is in a sustained form with no such
modulation.

String instruments in fact have a fairly broad bandwidth even on
sustained tones. Close spectral analysis reveals that drawing a bow
across a string in these instruments produces a plethora of
continuous, random frequencies in short, repetitive bursts around the
cneter frequency to which the string is tuned. This is responsible in
large part for the "thick", rich quality of the sound of strings,
quite in addition to the harmonicity of the string.

On reflection, this phenomenon is a natural consequence of the bow's
sticking to and pulling the string to increase its tension slightly
before it is released. Any experienced string player knows that one
can bow two adjacent strings tuned a perfect fifth apart and change
relative bow technique and pressure to detune the interval.

These "side" frequencies are present at all the harmonics above the
fundamental as well, since these latter are simply the harmonic
partials of the thick, broad spectra of the fundamental(s). We string
players find that we can judge pitch quite accurately in spite of
this relatively broad frequency bandwidth.

--- In tuning@y..., "J Gill" <JGill99@i...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > I'm posting this more as a springboard for debate than as an
actual
> > belief on my part. But it seems like something someone could
> > successfully argue, so I wonder what counterarguments there might
> be.
> >
> > The classical uncertainty principle is a mathematical fact
(unlike
> > the quantum-mechanical Heisenberg uncertainty principle, which is
> an
> > unexpected feature of nature). It derives from Fourier theory,
and
> > basically says that a musical note, played for time t, can have
its
> > frequency determined to no better than 2*pi/t (see
> > http://sepwww.stanford.edu/sep/prof/fgdp/c4/paper_html/node2.html
--
>
> > equation 1).
> >
> > Much has been made of the 3- and 4- cent deviations from JI in 72-
> tET
> > chords. At a typical musical frequency of 440Hz, 4 cents is a 1Hz
> > deviation. So the classical uncertainty principle would seem to
say
> > that, for frequency to be determined to better than this
accuracy,
> > the note would have to be played for 2*pi, or over 6 seconds!
> Clearly
> > most music has melodies and even chord changes that are much
faster
> > than this. Thus any attempt to say whether the chords were in JI
or
> > in 72-tET would be meaningless.
> >
> > Rebuttals?
>
>
> Paul,
>
> The gist of your proposed hypothesis above appears (to me) to imply
> that:
>
> (1) A listener's perception of the_difference_in the pitch of two
(or
> possibly more) simultaneously sounded sinusoidal (or possibly more
> spectrally complicated) tones is analogous to the process whereby a
> listener perceives the individual pitches of each of such
> (simultaneously sounded) individual tones; and, further that
>
> (2) The process of a listener's perception is limited by the same
> 2*PI cycle length sampling window length which a 2nd order bandpass
> filter network (and more sophisticated approaches utilizing Fourier
> Integrals, FFTs, and Z Transforms) require in order to provide
> results which can be considered reasonably valid.
>
>
> Regarding (2) above:
>
> Robert Walkers reply appears to explore some potentially
interesting
> possibilities for increasing the efficiency of certain "inorganic"
> processes in assessing pitch (or perhaps pitches) more rapidly than
> that of a time period of 2*PI cycles of the frequency.
>
> Some information which may be relevant to "organic" pitch
perception
> mechanisms follows. Harry F Olson's "Music, Physics, and
> Engineering", Dover, 1967, page 250, figure 7.6 depicts some data
> regarding the "duration of a tone in cycles required to ascribe a
> certain pitch (After Burck, Kotowski, and Lichte; von Bekesy and
> Turnbull)". From the text which describes figure 7.6:
>
> "The number of cycles required to ascribe a definite pitch to a
tone
> depends on the frequency, as shown in Fig. 7.6. It will be seen
that
> the number of cycles required to establish the pitch of a tone
> increases with the frequency. At 50 cycles it is 3 cycles,.."[LESS
> THAN 2*PI CYCLES] "at 400 cycles it is 7 cycles,.." [APPROX 2*PI
> CYCLES] "at 1,000 cycles it is 12 cycles, and at 8,000 cycles it is
> 145 cycles. It may be noted that the time required to ascribe a
> definite pitch does not vary a great deal from low to high
> frequencies. The average time is about 13 milliseconds."
>
> Figure 7.6 shows a roughly monotonic slope (in the_logarithm_of
> duration in cycles) between 50 CPS and 400 CPS. It is interesting
> that these findings imply that more may be going on in "organic"
> pitch perception processes than our best machinery may be able to,
in
> this respect,(reliably) resolve...
>
>
> Regarding (1) above:
>
> Can it really be said that a listener does not perceive the pitch
of
> a 440 CPS tone in, if you would, (2*PI)/440 seconds, or 14.3 mSEC,
> while perceiving the pitch of a second (simultaneously sounded)441
> CPS tone in, if you would, (2*PI)/441 seconds, or 14.25 mSEC?
>
> Can it really be said that any effects of the amplitude envelope
(in
> sound pressure level) varying at 1 CPS (due to any non-linear
> phenomena involved in the process which may generate difference
> frequencies) would not present themselves fully within 1/2 of one
> cycle of the 1 CPS difference frequency, or within 500 mSEC?
>
> It seems to me that the time required for a listener to
> resolve_differences_between frequencies may well be a different
> matter than the time required to resolve the individual frequencies
> (without broaching the potentially more complicated effects of such
> tones having harmonic as well as inharmonic spectral content).
> An "inorganic" example of this might be the small (but measurable)
> differential signal which exists between the amplitude envelopes
> (measured by instantaneous peak detection) of the outputs of two
> identical 2nd order bandpass filter networks both tuned to 440 CPS,
> where one filter is excited by a gated 440 CPS sinusoidal waveform,
> and the second filter is excited by a gated 441 CPS sinusoidal
> waveform. Such a differential signal would appear within one half-
> cycle of these_excitation_frequencies (1.135 mSEC), as opposed to
> having to wait 2*PI cycles (6.283 SEC) of the_difference_frequency,
> (which is a factor of 5535 times faster than the proposed delay).
>
>
> Regards, J Gill
>
>
> PS - Your "Forms of Tonality" paper is a thing of beauty. I really
> like form and quality of your graphics, and your text has, as you
had
> predicted, been helpful to me in taking in some of the ideas with
> which you have put forth. Thanks again for sending it along.
>
> PSS - Thanks for posting the link to Ernst Terhardt's web site, at:
> http://www.mmk.e-technik.tu-muenchen.de/persons/ter.html
> which has some_really_interesting stuff.

--- In tuning@y..., BobWendell@t... wrote:

> I, like most, I believe, can more easily detect an error of 5 cents
> when demonstrated as a harmonic interval, than as a melodic one in
> which the two tones are sounded sequentially. Nonetheless, a
meantone
> perfect fifth, for example (5.38 cents flat), does sound flat to me
> even MELODICALLY as long as the two tones are sounded in a context
> that introduces a low level of cognitive interference with the
> perception.

I'm not sure it's a matter of just a fifth taken in isolation. The 12-
et diatonic sounds different to my ears than the Pythagorean version,
and the meantone diatonic even more so, but those differences
accumulate.

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> Paul,
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > Because, for a given scale, 72 will allow you a greater number of
> > these consonant chords than JI.
>
> Maybe every composer doesn't need a greater number of consonant
> chords;

Of course!

> your implication is that fewer consonant chords is somehow
> inferior,

Come on Jon! My implication? Please. Look at what you wrote:

> Who cares??? If Daniel, or anyone else for that matter, couldn't
tell
> the difference, who cares whether it is JI or 72?

All I did was explain why _someone_ might care. I DID NOT imply that
one way was inferior!!!
>
> > Geez, when did I EVER suggest that a musician who did something
in
> > JI should have done it in 72tET instead?
>
> I would say the implication is there in your statements:

Well then you misread my statements.

> if the
> differences in audibility is "meaningless", and (as you state
above)
> there is some resource in 72tET in abundance over some other system
> such that you bring up the fact... why else?

To suggest to some, who may have otherwise closed their minds to the
possibility, why a tempered system might in fact be desirable for one
composer in one situation. I absolutely did not promote anything
as "inferior" or "superior", Jon. I simply want all the arguments to
be out in the open, and for all minds to be open. If you want to
present some relevant arguments, or simply ignore this issue, that
would be a lot better than putting positions into my mouth.

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> What one can do is to actually try it out and see if one can
> distinguish two notes 4 cents apart each of one second.
>
> Indeed, apparently some musicians can do the same
> with notes only 1 cent apart.

Apparently? Let's see even one bit of evidence for this.

> I think this matter of virtual pitch perception is rather
> beside the point.

It sure is -- not sure why one would bring it up.

> Here we are interested in very
> simple type of perception. E.g. perception of pitch
> of a sine wave would be fine.

Precisely!!
>
> I find recorder timbre is quite an easy one to use
> for distinguishing pitches on the soundcard, and it is
> pretty close to sine wave.
>
> I get the impression nobody quite knows how we are able to
> perceive the pitch of, say, a sine wave.

That's not true at all. I quote Terhardt: it's

'evident that the pitch of sine tones with high probability is
a "place pitch", i.e., is dependent on the place of maximal
excitation of the cochlear partition, and such eventually is a result
of peripheral auditory Fourier analysis.'

Note that he's very careful and says "with high probability" -- if
you read through his references, you'll see that the evidence is
overwhelmingly in favor of this theory.
>
> Hope also no-one gets discouraged from developing their
> pitch sensitivity further

Agreed!

> by this argument as it isn't
> a valid one.

Jay Gill's report seems to suggest that you're right. However, in
some contexts, i.e., a rapidly changing musical stimulus, perhaps
with timbral and/or background noise, and/or inharmonic partials, the
basic idea behind the argument may still be relevant.

--- In tuning@y..., BobWendell@t... wrote:

> Jon, is it not perfectly legitimate and even important that we
spell
> out the differences and relative properties, leaving the choice up
to
> the creative desires of the musician/composer? How does doing this
> automatically imply all the things you say Paul has implied?

Thank you Bob, for the outsider perspective. It's always nice to be
reassured that I haven't gone crazy.

I agree, but was holding this to the simple case of a minimal pitch
comparison of two tones in order to stay as relevant as possible to
the basic argument.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > I, like most, I believe, can more easily detect an error of 5
cents
> > when demonstrated as a harmonic interval, than as a melodic one
in
> > which the two tones are sounded sequentially. Nonetheless, a
> meantone
> > perfect fifth, for example (5.38 cents flat), does sound flat to
me
> > even MELODICALLY as long as the two tones are sounded in a
context
> > that introduces a low level of cognitive interference with the
> > perception.
>
> I'm not sure it's a matter of just a fifth taken in isolation. The
12-
> et diatonic sounds different to my ears than the Pythagorean
version,
> and the meantone diatonic even more so, but those differences
> accumulate.

I'm reposting this for a minor correction for the sake of clarity and
simply because no one seems to have noticed it yet. It is a challenge
to the fundamental premise of the argument Paul put out for us to
chew on.

--- In tuning@y..., BobWendell@t... wrote:
> I just went back again to the paper from which this discussion took
> its basis and keyed in on something I'd overlooked. The whole
thesis
> of frequency uncertainty is based on the SPECTRAL BANDWIDTH of
tones
> that are amplitude modulated in ways that can be described by
attack
> and decay envelopes plotted against time. Clearly, many if not most
> musical tones, especially short ones that are percussively or
> "pluckingly" produced, have significant frequency bandwidth
> broadening due to attack and decay envelopes.
>
> The argument that Paul has cleverly posited is based on the
> assumption that spectral bandwidth is synonymous with uncertainty
> concerning the frequency modulated by the attack and decay
envelopes
> in question. This assumption is fundmentally flawed. Bandwidth very
> frequently has NOTHING to do with uncertainty as to the frequency
> being modulated to produce that bandwidth. The bandwidth introduced
> by such amplitude modulations is INHERENTLY SYMMETRICAL around the
> base frequency. Psychoacoustically the center frequency is still
> quite perceptibly the same as it is in a sustained form with no
such
> modulation.
>
> String instruments in fact have a fairly broad bandwidth even on
> sustained tones. Close spectral analysis reveals that drawing a bow
> across a string in these instruments produces a plethora of
> continuous, random frequencies in short, repetitive bursts around
the
> cneter frequency to which the string is tuned. This is responsible
in
> large part for the "thick", rich quality of the sound of strings,
> quite in addition to the harmonicity of the string.
>
> On reflection, this phenomenon is a natural consequence of the
bow's
> sticking to and pulling the string to increase its tension slightly
> before it is released. Any experienced string player knows that one
> can bow two adjacent strings tuned a perfect fifth apart and change
> relative bow technique and pressure to detune the interval.
>
> The "side" frequencies present in bowed string timbres are present
at all the harmonics above the fundamental as well, since these
latter are simply the harmonic partials of the thick, broad spectra
of the fundamental(s). We string players find that we can judge pitch
quite accurately in spite of this relatively broad frequency
bandwidth.
>
>

--- In tuning@y..., BobWendell@t... wrote:

> In this context, though, I must say I have heard music samples
> referred to here and there in this list claiming to use tuning
> systems the intonational accuracies of which were grossly violated
by
> singers and/or instrumentalists involved in the performance.

Go, Bob! How very true. See, this whole thread is really trying to
make the point that it's useless to be religious about tuning to a
level of precision that one can't even hear. Music should be about
_sound_, not about numbers on a page (and though there's nothing
wrong with plunging into the latter if it's in the service of the
former, I have a big problem with the reverse). And when those who
espouse tuning religions which assert their superiority on this or
that mythological basis to marginally different alternatives then
produce music which embarassingly fails to adhere to the proposed
tuning, well you just kinda hafta wonda. I happen to be a
professional musician whose ear has received particular recognition
throughout my life -- thus my interest in tuning is no coincidence.
Others with other talents have just as much of a right to pursue
these issues as I do. But where they cease to have any relationship
with the audible, they cease to have any importance to me, and as far
as I can see, to music. Music is magical to me. But unless someone
proves otherwise, I have no reason to believe that it has any effect
beyond its audible content. If someone wrote a synthesizer piece in
JI, but all their pitch bends were off by a factor of two, would
their "intent" somehow mystically be communicated to me, even if the
piece were transmitted over the internet or on a CD? Well, as a
person living in the real world, making music for real people who
_listen_ and don't "intuit intent", I'll have to say that my
experience favors a "no" answer. I can't tell you how many times I
walked on stage to pick up one instrument or another, greeted with
extremely sceptical glares based on my appearance (I looked very
young a few years ago, and sometimes I'm dressed in business attire,
or the extreme opposite), only to be told, after my performance, that
I'd greatly exceeded expectations. _That's_ meaningful to me. I don't
give a d**n how nice a piece of music might look on a piece of paper -
- whether JI ratios or 12-tone serialism. It's when eyes are shut and
sound is on that the true test occurs. Otherwise, you'd be better off
writing poetry or painting pictures, as far as I'm concerned.

***ramble mode off***

--- In tuning@y..., BobWendell@t... wrote:

> Bandwidth very
> frequently has NOTHING to do with uncertainty as to the frequency
> being modulated to produce that bandwidth.

This seems to be correct. But where might it have SOMETHING to do
with uncertainty?

> The bandwidth introduced
> by such amplitude modulations is INHERENTLY SYMMETRICAL around the
> base frequency.

But what if, rather than an amplitude modulation from off to on to
off again, we're talking about a frequency modulation from one pitch
to another pitch to another pitch?
>
> These "side" frequencies are present at all the harmonics above the
> fundamental as well, since these latter are simply the harmonic
> partials of the thick, broad spectra of the fundamental(s). We
string
> players find that we can judge pitch quite accurately in spite of
> this relatively broad frequency bandwidth.

Perhaps ironically, it's far more difficult to judge the pitch of
pure sine waves with as much, or even quite a bit less, accuracy. But
perhaps we should start a psychoacoustics list to discuss this
further, before it becomes misinterpreted as a discussion
of "superiority" and "inferiority"!

Good belly laugh, Paul!!! I'm glad I somehow inspired that.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > In this context, though, I must say I have heard music samples
> > referred to here and there in this list claiming to use tuning
> > systems the intonational accuracies of which were grossly
violated
> by
> > singers and/or instrumentalists involved in the performance.
>
> Go, Bob! How very true. See, this whole thread is really trying to
> make the point that it's useless to be religious about tuning to a
> level of precision that one can't even hear. Music should be about
> _sound_, not about numbers on a page (and though there's nothing
> wrong with plunging into the latter if it's in the service of the
> former, I have a big problem with the reverse). And when those who
> espouse tuning religions which assert their superiority on this or
> that mythological basis to marginally different alternatives then
> produce music which embarassingly fails to adhere to the proposed
> tuning, well you just kinda hafta wonda. I happen to be a
> professional musician whose ear has received particular recognition
> throughout my life -- thus my interest in tuning is no coincidence.
> Others with other talents have just as much of a right to pursue
> these issues as I do. But where they cease to have any relationship
> with the audible, they cease to have any importance to me, and as
far
> as I can see, to music. Music is magical to me. But unless someone
> proves otherwise, I have no reason to believe that it has any
effect
> beyond its audible content. If someone wrote a synthesizer piece in
> JI, but all their pitch bends were off by a factor of two, would
> their "intent" somehow mystically be communicated to me, even if
the
> piece were transmitted over the internet or on a CD? Well, as a
> person living in the real world, making music for real people who
> _listen_ and don't "intuit intent", I'll have to say that my
> experience favors a "no" answer. I can't tell you how many times I
> walked on stage to pick up one instrument or another, greeted with
> extremely sceptical glares based on my appearance (I looked very
> young a few years ago, and sometimes I'm dressed in business
attire,
> or the extreme opposite), only to be told, after my performance,
that
> I'd greatly exceeded expectations. _That's_ meaningful to me. I
don't
> give a d**n how nice a piece of music might look on a piece of
paper -
> - whether JI ratios or 12-tone serialism. It's when eyes are shut
and
> sound is on that the true test occurs. Otherwise, you'd be better
off
> writing poetry or painting pictures, as far as I'm concerned.
>
> ***ramble mode off***

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
Bob:
> > Bandwidth very
> > frequently has NOTHING to do with uncertainty as to the frequency
> > being modulated to produce that bandwidth.
>

Paul:
> This seems to be correct. But where might it have SOMETHING to do
> with uncertainty?
>
Bob answers now:
Only if the modulation were very much more complex, wideband, and not
simply attack and decay envelopes, which are of relatively low
frequency. The latter only add significant bandwidth because of their
transient nature and shapes, which generate lots of harmonic content
in the modulating signal, widening it's bandwidth.

Bob:
> > The bandwidth introduced
> > by such amplitude modulations is INHERENTLY SYMMETRICAL around
the
> > base frequency.
>
Paul:
> But what if, rather than an amplitude modulation from off to on to
> off again, we're talking about a frequency modulation from one
pitch
> to another pitch to another pitch?
> >
Bob answers now:
This lies well outside the scope of the original discussion and the
mathematics behind it, all of which was predicated on the amplitude
modulation implicit in rise time (attack envelope) and decay time
(decay envelope).

Bob:
> > These "side" frequencies are present at all the harmonics above
the
> > fundamental as well, since these latter are simply the harmonic
> > partials of the thick, broad spectra of the fundamental(s). We
> string
> > players find that we can judge pitch quite accurately in spite of
> > this relatively broad frequency bandwidth.
>
Paul:
> Perhaps ironically, it's far more difficult to judge the pitch of
> pure sine waves with as much, or even quite a bit less, accuracy.
But
> perhaps we should start a psychoacoustics list to discuss this
> further, before it becomes misinterpreted as a discussion
> of "superiority" and "inferiority"!

Bob answers now:
I hate to splinter off again into a side list. It's cumbersome, but I
understand your point, especially regarding the issue of
"inferiority/superiority".

--- In tuning@y..., BobWendell@t... wrote:

> Paul:
> > But what if, rather than an amplitude modulation from off to on
to
> > off again, we're talking about a frequency modulation from one
> pitch
> > to another pitch to another pitch?
> > >
> Bob answers now:
> This lies well outside the scope of the original discussion

Not at all! Seems like an evasive maneuver there, Bob.

> and the
> mathematics behind it, all of which was predicated on the amplitude
> modulation implicit in rise time (attack envelope) and decay time
> (decay envelope).

Does the classical uncertainty principle really fail to apply in
other types of circumstances?

> Bob answers now:
> I hate to splinter off again into a side list. It's cumbersome, but
I
> understand your point, especially regarding the issue of
> "inferiority/superiority".

Perhaps we can discuss psychoacoustics on the tuning-math list,
because it does tend to get somewhat mathematical . . . ? You seem to
have great knowledge here and I'd love to continue . . .

--- In tuning@y..., BobWendell@t... wrote:
>
> This only shows limitations in the perceptual apparatus and not one
> of mathematical principle. In the same post, I clearly implied that
I
> could easily distinguish a 5 cent error even melodically in much
less
> than a second, so that already takes us beyond what this
mathematical
> argument would allow, considering the shortness of the time frame.

Does it? Are you using sine waves? If you're using complex tones at
440Hz, then the Nth harmonic would allow you to acheive N times more
accuracy, just using the same argument.

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

> Go, Bob! How very true. See, this whole thread is really trying to
> make the point that it's useless to be religious about tuning to a
> level of precision that one can't even hear. Music should be about
> _sound_, not about numbers on a page (and though there's nothing
> wrong with plunging into the latter if it's in the service of the
> former, I have a big problem with the reverse).

I'm bummed--does this mean you don't want to hear about the wonderful

29/1848 < 134/8539 < 105/6691

generator? It maps to primes by 101, -553, -76, 93; and we could ask
the religious question of whether errors of .00073, .00787, -.00052,
-.01217, (and -.00769 for the 13) would be good enough for Harry
Partch.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
> >
> > This only shows limitations in the perceptual apparatus and not
one
> > of mathematical principle. In the same post, I clearly implied
that
> I
> > could easily distinguish a 5 cent error even melodically in much
> less
> > than a second, so that already takes us beyond what this
> mathematical
> > argument would allow, considering the shortness of the time frame.
>
> Does it? Are you using sine waves? If you're using complex tones at
> 440Hz, then the Nth harmonic would allow you to acheive N times
more
> accuracy, just using the same argument.

Good point, Paul. Missed that one, but in my other posts have
indicated that the basic premise is flawed.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Go, Bob! How very true. See, this whole thread is really trying
to
> > make the point that it's useless to be religious about tuning to
a
> > level of precision that one can't even hear. Music should be
about
> > _sound_, not about numbers on a page (and though there's nothing
> > wrong with plunging into the latter if it's in the service of the
> > former, I have a big problem with the reverse).
>
> I'm bummed--does this mean you don't want to hear about the
wonderful
>
> 29/1848 < 134/8539 < 105/6691
>
> generator? It maps to primes by 101, -553, -76, 93; and we could
ask
> the religious question of whether errors of .00073, .00787, -
.00052,
> -.01217, (and -.00769 for the 13) would be good enough for Harry
> Partch.

Well, since you won't get any extra consonances by tuning Partch's
scales in this way, nor will this temperament make for an
intelligible keyboard mapping, I'm not too interested . . . isn't
this what you yourself would call a "fetish"? . . . but to the extent
that one might be curious in the numbers for their own sake, I see
that 1848, 6691, and 8539 all appear here:

http://library.wustl.edu/~manynote/consist2.txt

which document is explained here:

http://library.wustl.edu/~manynote/music.html

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > Paul:
> > > But what if, rather than an amplitude modulation from off to on
> to
> > > off again, we're talking about a frequency modulation from one
> > pitch
> > > to another pitch to another pitch?
> > > >
> > Bob answers now:
> > This lies well outside the scope of the original discussion
>
> Not at all! Seems like an evasive maneuver there, Bob.
>
Bob answers again:
The original argument is based on amplitude modulation of fixed-
frequency tones in the form of attack and decay envelopes (rise and
decay times). This increases the bandwidth of the sounded tone, but
does not obscure its pitch, that is, does NOT extend the imprecision
of pitch perception to the bandwidth this modulation generates.

Frequency modulation (FM) is a whole different ballgame and has
nothing to do with the original premise of this argument. That's all
I'm saying. Not being evasive at all! It's a competely different,
unrelated subject, unless you want to include anything at all having
to do with pitch perception as on topic.

Bob:
> > and the
> > mathematics behind it, all of which was predicated on the
amplitude
> > modulation implicit in rise time (attack envelope) and decay time
> > (decay envelope).
>

Paul:
> Does the classical uncertainty principle really fail to apply in
> other types of circumstances?
>
> > Bob answers now:
I'm sure it's more general than that, but don't think this is a
cogent application of its implications for the reasons already given.

--- In tuning@y..., BobWendell@t... wrote:
>
> Frequency modulation (FM) is a whole different ballgame and has
> nothing to do with the original premise of this argument.

I said, what if amplitude doesn't go from off to on to off again, but
instead the frequency goes from one value to another value to another
value. I don't see why this has any less or more to do with the
original premise, considering how various musical instruments produce
musical passages.

> That's all
> I'm saying. Not being evasive at all!

It sure seems evasive to me!

> It's a competely different,
> unrelated subject,

??? Look at my original post again. Did I say anywhere that the tone,
whose duration I was saying was typically less than 6 seconds, needed
to be preceded and followed by silence, rather than by other pitches?
I was talking about pitches in a musical context, where both
situations are quite likely.
>
> Bob:
> > > and the
> > > mathematics behind it, all of which was predicated on the
> amplitude
> > > modulation implicit in rise time (attack envelope) and decay
time
> > > (decay envelope).
> >
>
> Paul:
> > Does the classical uncertainty principle really fail to apply in
> > other types of circumstances?
> >
> > > Bob answers now:
> I'm sure it's more general than that, but don't think this is a
> cogent application of its implications for the reasons already >
given.

You've given no such reasons whatsoever.

--- In tuning@y..., BobWendell@t... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., BobWendell@t... wrote:

> <SNIP>
>
> Bob:
> > > The bandwidth introduced
> > > by such amplitude modulations is INHERENTLY SYMMETRICAL around
> the
> > > base frequency.
> >
> Paul:
> > But what if, rather than an amplitude modulation from off to on
to
> > off again, we're talking about a frequency modulation from one
> pitch
> > to another pitch to another pitch?
> > >
> Bob answers now:
> This lies well outside the scope of the original discussion and the
> mathematics behind it, all of which was predicated on the amplitude
> modulation implicit in rise time (attack envelope) and decay time
> (decay envelope).

J Gill:

In a case where the "modulation index" of frequency modulation of a
sinusoidal carrier wave(Wc) by a sinusoidal modulating wave(Wm),
which is equal to dWc/wm (d=delta), is less than a value of unity
(narrowband FM) the first-order sidebands (above and below the
carrier frequency) predominate in a frequency modulated spectrum,
resulting in a_magnitude_spectrum which is very similar to that of an
amplitude modulated spectrum (where the energy is distributed at the
carrier frequency as well as at an upper and a lower sideband).

The phase of this first lower sideband (and all odd numbered lower
sidebands) is, however, 180 degrees out of phase with that of each of
the odd numbered upper sidebands (as opposed to the equal phase of
all of the upper and the sidebands resulting from amplitude
modulation).

>
> Bob:
> > > These "side" frequencies are present at all the harmonics above
> the
> > > fundamental as well, since these latter are simply the harmonic
> > > partials of the thick, broad spectra of the fundamental(s). We
> > string
> > > players find that we can judge pitch quite accurately in spite
of
> > > this relatively broad frequency bandwidth.
> >
> Paul:
> > Perhaps ironically, it's far more difficult to judge the pitch of
> > pure sine waves with as much, or even quite a bit less, accuracy.

J Gill:

Paul, could you more explicitly describe the situation in which you
have framed this "line of thought", that is - the frequency (or
frequencies) that are present and relevant (you stated to your
response to me that it did not include simultaneous energy at 440 CPS
as well as 441 CPS). What _does_it include? Are we talking about sine
waves only, or the presence of harmonic energy at multiples of the
fundamental(s), as well?

Is the pitch "uncertainty" which you address a difference between a
tone which_is_being sounded and a tone which_is not_being sounded at
that time (such as a 72-tEt pitch where the "mind" might
be "expecting" a JI pitch, instead), or is it a difference between
two tones which both_are_being sounded simultaneously? If your
situation constitutes the former, then how does the "mind" know what
frequency it "should be" hearing (aside from the case of that
anticipated pitch being an integer multiple of a lower frequency
fundamental frequency which is simultaneously being sounded).

Curiously, J Gill

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

> Well, since you won't get any extra consonances by tuning Partch's
> scales in this way, nor will this temperament make for an
> intelligible keyboard mapping, I'm not too interested . . . isn't
> this what you yourself would call a "fetish"?

I'm a number theorist--I like number properties for their own sake.

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:

/tuning/topicId_29265.html#29312

> Paul,
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > However, I'd be happy to apply the argument to claim that, if
Kraig were to play an 11-limit Utonal chord, one (even Daniel Wolf)
could not tell whether it was in JI or in 72-tET, unless the chord
were held for a long time (on the order of 2*pi seconds).
>
> Who cares??? If Daniel, or anyone else for that matter, couldn't
tell the difference, who cares whether it is JI or 72?
>
> Once again, the realm of musical choice, taste, and philosophy is
> asked to take a back seat to 'arguements'. I resist - constantly -
to respond to these kind of 'debates' (which aren't really debates,
but
> one person trying to find more 'proof' of one side being superior
> over another), because I'll end up sounding like someone I don't
like
> in the least. But you can write all you want about why someone who
> did something in JI could have done it -- what: better? cheaper?
> lower fat content? -- in 72tET, and it will ALWAYS miss the point:
> that was their choice, and there is *nothing* wrong with it.
>

Well... maybe this is answered or elaborated further down in the
thread, but certainly in the case of 72-tET it makes a *big*
difference whether one were to use that system rather than Just
Intonation. If it really is true (as it apparently is) that one can
come excessively close to JI using 72-tET than that is an *extremely*
important conclusion... since the *notation* for 72 equal is *so*
much easier to handle than most Just Intonation notation systems...
(Ben Johnston comes immediately to mind...)

_________ _______ __________
Joseph Pehrson

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

/tuning/topicId_29265.html#29318

>
> > because I'll end up sounding like someone I don't like
> > in the least. But you can write all you want about why someone
who did something in JI could have done it -- what: better? cheaper?
> > lower fat content? -- in 72tET, and it will ALWAYS miss the
point: that was their choice, and there is *nothing* wrong with it.
>
> Geez, when did I EVER suggest that a musician who did something in
JI should have done it in 72tET instead?

I don't believe you did, Paul, but frankly *I* would say that. Why??
Because 72-tET is so much easier to notate and to perform!

________ _________ _______
Joseph Pehrson

I have an old habit of assuming people can hear more than they
usually can even though my own experience indicates that I hear
intonationally better than 99+% of the musical world. I tend, rather
than simply accepting that, to project my own ability to hear on
everyone else. That is typical of us humans. We tend to tacitly,
unconsciously assume that others' experience parallels our own.

This having been said, the difference between 72-tET and JI is for me
musically small enough (not mathematically, Gene) to become
practically insignificant, and the above-cited experience would
indicate that the difference would be even less significant for the
everwhelming majority of musically-involved humanity.

I was looking for the lowest possble number of equal steps per octave
that would serve as a convenient but still satisfactory conceptual
model for JI. Paul Erlich suggested 72-tET. I have looked into its
properties thoroughly and feel it is quite satisfactory.

In addition to many other useful properties, it offers BY FAR the
closest rational approximation to the generator we have dubbed the
secor (116.7156 cents), after its author, of any ET until you are way
up into the hundreds and the step sizes themselves become so small
that the exercise of optimization begins to become meaningless.

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_29265.html#29318
>
> >
> > > because I'll end up sounding like someone I don't like
> > > in the least. But you can write all you want about why someone
> who did something in JI could have done it -- what: better?
cheaper?
> > > lower fat content? -- in 72tET, and it will ALWAYS miss the
> point: that was their choice, and there is *nothing* wrong with it.
> >
> > Geez, when did I EVER suggest that a musician who did something
in
> JI should have done it in 72tET instead?
>
>
> I don't believe you did, Paul, but frankly *I* would say that.
Why??
> Because 72-tET is so much easier to notate and to perform!
>
> ________ _________ _______
> Joseph Pehrson

Well, Paul, I'm baffled. I don't want to start quoting from my
previous posts in this thread as a rebuttal, so I'm at a bit of a
loss. Maybe I can summarize succinctly:

The paper upon which you predicated your initial question uses a
mathematical argument that, if I understand it correctly, assumes
pitch uncertainty as implicit in the bandwidth around a fundamental
of a stable, FIXED frequency introduced by the time rise and decay
present in any practical musical tone. What does that have to do with
FM? The delta omega term does NOT refer to absolute frequency, but a
RANGE of frequencies around the original fundamental and its partials
called BANDWIDTH.

Frequency modulation as a technology for generating synth sounds is
not in that picture and in any case the ultimate product of such
synthesis is still a fundamental and a set of partials, whether
harmonic or not. If we consider VIBRATO as very low frequency FM of
the musical tone, then that is a whole separate issue and the pitch
uncertainty that might result from a very wide Frequency modulation
in that case is a result of the width of the vibrato. This latter
phenomenon is one that is arbitrarily introduced by the musician's
choice and is not inherent in the attack and decay envelopes.

I simply posit that the simple case (the one advanced in the paper)
of a tone with a fundamental at a single frequency does NOT become
mildly indeterminant because of the bandwidth introduced by attack
and decay envelopes. That is all.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
> >
> > Frequency modulation (FM) is a whole different ballgame and has
> > nothing to do with the original premise of this argument.
>
> I said, what if amplitude doesn't go from off to on to off again,
but
> instead the frequency goes from one value to another value to
another
> value. I don't see why this has any less or more to do with the
> original premise, considering how various musical instruments
produce
> musical passages.
>
> > That's all
> > I'm saying. Not being evasive at all!
>
> It sure seems evasive to me!
>
> > It's a competely different,
> > unrelated subject,
>
> ??? Look at my original post again. Did I say anywhere that the
tone,
> whose duration I was saying was typically less than 6 seconds,
needed
> to be preceded and followed by silence, rather than by other
pitches?
> I was talking about pitches in a musical context, where both
> situations are quite likely.
> >
> > Bob:
> > > > and the
> > > > mathematics behind it, all of which was predicated on the
> > amplitude
> > > > modulation implicit in rise time (attack envelope) and decay
> time
> > > > (decay envelope).
> > >
> >
> > Paul:
> > > Does the classical uncertainty principle really fail to apply
in
> > > other types of circumstances?
> > >
> > > > Bob answers now:
> > I'm sure it's more general than that, but don't think this is a
> > cogent application of its implications for the reasons already >
> given.
>
> You've given no such reasons whatsoever.

[Joseph Pehrson wrote:]
>Well... maybe this is answered or elaborated further down in the
>thread, but certainly in the case of 72-tET it makes a *big*
>difference whether one were to use that system rather than Just
>Intonation. If it really is true (as it apparently is) that one can
>come excessively close to JI using 72-tET than that is an *extremely*
>important conclusion... since the *notation* for 72 equal is *so*
>much easier to handle than most Just Intonation notation systems...
>(Ben Johnston comes immediately to mind...)

I missed the actual question, but I'm going to sound, again, my note(s)
of caution regarding 72-tET.

. It is not anywhere near "excessively close to JI". Its errors
through the 11-limit are acceptable to my ear, much better than
12-tET, but 4 cents is far from trivial to many ears.

. More importantly, IMHO, is the "straight jacket" it imposes with
regard to the syntonic comma. Any music with a normal diatonic
progression C-A-D-G-C has either got to drift or shift a comma all
at once (howbeit a reduced comma at 16.67 cents).

I know that many poo-poo this second objection (Paul E, are you reading
this?), and I freely acknowledge that it reflects my own approach to
tuning, an approach best suited to instruments whose intonation is
continuously variable.

The notation for 72-tET is conveniently reducible into six deviations
from 12-tET (one being blank, no deviation). It is compact. Still, I
would prefer that a Reinhard type notation (cents deviation from 12-tET)
be adopted in place of 72-tET, because it allows commas to be taken
on the fly, a few cents at a time, instead of suddenly, painfully.

JdL

--- In tuning@y..., BobWendell@t... wrote:
> Well, Paul, I'm baffled. I don't want to start quoting from my
> previous posts in this thread as a rebuttal, so I'm at a bit of a
> loss. Maybe I can summarize succinctly:
>
> The paper upon which you predicated your initial question

I didn't predicate it on any paper. I may have given a web address to
one example where the classical UP was discussed -- that's all.

Now let me repeat:

> > I said, what if amplitude doesn't go from off to on to off again,
> but
> > instead the frequency goes from one value to another value to
> another
> > value. I don't see why this has any less or more to do with the
> > original premise, considering how various musical instruments
> produce
> > musical passages.

You said that, in the case where the amplitude goes from off to on to
off again, one could simply latch onto the center of the Fourier
transform and get a precise frequency reading. OK. But what if the
frequency goes from one value to the value in question to another
value (say the same as the first value). Surely the classical UP
still applies to the middle frequency. But is the _center_ of the
Fourier transform still going to give you the value you're looking
for? That's what I was asking.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Joseph Pehrson wrote:]
> >Well... maybe this is answered or elaborated further down in the
> >thread, but certainly in the case of 72-tET it makes a *big*
> >difference whether one were to use that system rather than Just
> >Intonation. If it really is true (as it apparently is) that one
can
> >come excessively close to JI using 72-tET than that is an
*extremely*
> >important conclusion... since the *notation* for 72 equal is *so*
> >much easier to handle than most Just Intonation notation
systems...
> >(Ben Johnston comes immediately to mind...)
>
> I missed the actual question, but I'm going to sound, again, my note
(s)
> of caution regarding 72-tET.
>
> . It is not anywhere near "excessively close to JI". Its errors
> through the 11-limit are acceptable to my ear, much better than
> 12-tET, but 4 cents is far from trivial to many ears.
>
> . More importantly, IMHO, is the "straight jacket" it imposes
with
> regard to the syntonic comma. Any music with a normal diatonic
> progression C-A-D-G-C has either got to drift or shift a comma
all
> at once (howbeit a reduced comma at 16.67 cents).
>
> I know that many poo-poo this second objection (Paul E, are you
reading
> this?),

Hi John,

Well, I've probably discussed and warned and whined and complained
about the syntonic comma problem more than anyone -- but you have to
remember that anyone composing in strict JI has already had to deal
with this problem, or avoid it completely, somehow . . . so if the
decision is between 72-tET and strict JI, this does not become a
deciding factor.

My two cents: If we're talking about tuning historical performances
with adaptive JI, I agree totally with John's perspective.
Personally, my interest in 72-tET is for NEW music, precisely because
it models quite well the properties native to JI, but within a
manageably finite set of equal steps per octave. VERY convenient.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> > [Joseph Pehrson wrote:]
> > >Well... maybe this is answered or elaborated further down in the
> > >thread, but certainly in the case of 72-tET it makes a *big*
> > >difference whether one were to use that system rather than Just
> > >Intonation. If it really is true (as it apparently is) that one
> can
> > >come excessively close to JI using 72-tET than that is an
> *extremely*
> > >important conclusion... since the *notation* for 72 equal is
*so*
> > >much easier to handle than most Just Intonation notation
> systems...
> > >(Ben Johnston comes immediately to mind...)
> >
> > I missed the actual question, but I'm going to sound, again, my
note
> (s)
> > of caution regarding 72-tET.
> >
> > . It is not anywhere near "excessively close to JI". Its
errors
> > through the 11-limit are acceptable to my ear, much better
than
> > 12-tET, but 4 cents is far from trivial to many ears.
> >
> > . More importantly, IMHO, is the "straight jacket" it imposes
> with
> > regard to the syntonic comma. Any music with a normal
diatonic
> > progression C-A-D-G-C has either got to drift or shift a
comma
> all
> > at once (howbeit a reduced comma at 16.67 cents).
> >
> > I know that many poo-poo this second objection (Paul E, are you
> reading
> > this?),
>
> Hi John,
>
> Well, I've probably discussed and warned and whined and complained
> about the syntonic comma problem more than anyone -- but you have
to
> remember that anyone composing in strict JI has already had to deal
> with this problem, or avoid it completely, somehow . . . so if the
> decision is between 72-tET and strict JI, this does not become a
> deciding factor.

--- In tuning@y..., BobWendell@t... wrote:
> My two cents: If we're talking about tuning historical performances
> with adaptive JI, I agree totally with John's perspective.

As do I.

> Personally, my interest in 72-tET is for NEW music, precisely
because
> it models quite well the properties native to JI, but within a
> manageably finite set of equal steps per octave. VERY convenient.

Even bigger conveniences of 72-tET are the facts that it contains the
standard 12-tET as its 3-limit, "Pythagorean" subset, and that each
new prime up through 11 involves a unique "accidental" or alteration
from the 12-tET grid. The main difference between JI and 72-tET that
has been brought up is the fact that 225:224 vanishes in 72-tET, but
it's 7.7 cents in JI. The vanishing gives 72-tET many advantages, as
in the extra consonant chords in the Fokker-Lumma 12-tone scale and
of course the MIRACLE scales. It can be a disadvantage to those, like
Rami Vitale, who insist on having 15:14 and 16:15 as different-sized
intervals.

[I wrote:]
>>I know that many poo-poo this second objection (Paul E, are you
>>reading this?),

[Paul wrote:]
>Hi John,

>Well, I've probably discussed and warned and whined and complained
>about the syntonic comma problem more than anyone --

Yes, I know you have. I was being bad on account of the last time I
wrote on this theme, you came back and said, don't worry, [whomever]
isn't going to hit this problem.

[Paul:]
>but you have to remember that anyone composing in strict JI has already
>had to deal with this problem, or avoid it completely, somehow . . . so
>if the decision is between 72-tET and strict JI, this does not become a
>deciding factor.

Quite true. My complaint is that 72-tET does not address this serious
problem of strict JI. The solution is at hand, but 72-tET sidesteps it!

JdL

--- In tuning@y..., BobWendell@t... wrote:

> This having been said, the difference between 72-tET and JI is for
me
> musically small enough (not mathematically, Gene) to become
> practically insignificant, and the above-cited experience would
> indicate that the difference would be even less significant for the
> everwhelming majority of musically-involved humanity.

I would say that for many people you will have reached that point
already with the 53-et, and for many others the 31-et will sound
better to them, JI or not. What is significant depends on what you
are counting as significant; of course with enough steps the question
does finally become moot.

> I was looking for the lowest possble number of equal steps per
octave
> that would serve as a convenient but still satisfactory conceptual
> model for JI. Paul Erlich suggested 72-tET.

And I would suggest the problem is not well-defined, and that you
could equally well argue that 53 or 171 is the correct answer.

> In addition to many other useful properties, it offers BY FAR the
> closest rational approximation to the generator we have dubbed the
> secor (116.7156 cents), after its author, of any ET until you are
way
> up into the hundreds and the step sizes themselves become so small
> that the exercise of optimization begins to become meaningless.

And 53 does excellently well for the Orwell (31+22) and Schismatic
(41+12) generators, and so forth. Trying to decide which of these
generators, or which et, is "the best" is a but like trying to decide
if the "best" place to take a vacation is Italy, Tahiti or Orlando,
Florida; the various systems are simply structured quite differently
and do different things.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> . It is not anywhere near "excessively close to JI". Its errors
> through the 11-limit are acceptable to my ear, much better than
> 12-tET, but 4 cents is far from trivial to many ears.

Agreed.

> . More importantly, IMHO, is the "straight jacket" it imposes
with
> regard to the syntonic comma. Any music with a normal diatonic
> progression C-A-D-G-C has either got to drift or shift a comma
all
> at once (howbeit a reduced comma at 16.67 cents).

Joseph wanted something close to JI; if he asked for a good meantone
system the answer would have been different. It's simply a feature,
not a bug, of JI that your "normal diatonic progression" transposes
down a comma--5/3 * 2/3 * 4/3 * 2/3 = 80/81 is a fact of arithmetic.

>Still, I
> would prefer that a Reinhard type notation (cents deviation from 12-
tET)
> be adopted in place of 72-tET, because it allows commas to be taken
> on the fly, a few cents at a time, instead of suddenly, painfully.

The easy way to do that is meantone, of course.

Yes, I've certainly noticed that the intervals in 72-tET repeat
several times starting at the twelfth harmonic (e.g., 12:13 and 13:14
are both 8 steps, then 14:15 and 15:16 are both 7, then 3 or so
intervals at 6, etc.). The neat thing is how nicely this temperament
tracks all the way up these increasingly smaller intervals even if in
this stepwise, discrete way.

From a western compositional viewpoint, I think it's a small price to
pay for its advantages. I see no reason to use it for Indian
classical music, though, since that system is already well-defined
and quite comfortably finite.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
> > My two cents: If we're talking about tuning historical
performances
> > with adaptive JI, I agree totally with John's perspective.
>
> As do I.
>
> > Personally, my interest in 72-tET is for NEW music, precisely
> because
> > it models quite well the properties native to JI, but within a
> > manageably finite set of equal steps per octave. VERY convenient.
>
> Even bigger conveniences of 72-tET are the facts that it contains
the
> standard 12-tET as its 3-limit, "Pythagorean" subset, and that each
> new prime up through 11 involves a unique "accidental" or
alteration
> from the 12-tET grid. The main difference between JI and 72-tET
that
> has been brought up is the fact that 225:224 vanishes in 72-tET,
but
> it's 7.7 cents in JI. The vanishing gives 72-tET many advantages,
as
> in the extra consonant chords in the Fokker-Lumma 12-tone scale and
> of course the MIRACLE scales. It can be a disadvantage to those,
like
> Rami Vitale, who insist on having 15:14 and 16:15 as different-
sized
> intervals.

Unless we're recreating (i.e., retuning) historical music, then why
regard this as a "problem" of JI rather than as a PROPERTY INDIGENOUS
TO IT AND TO BE COMPOSITIONALLY EXPLOITED???????

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [I wrote:]
> >>I know that many poo-poo this second objection (Paul E, are you
> >>reading this?),
>
> [Paul wrote:]
> >Hi John,
>
> >Well, I've probably discussed and warned and whined and complained
> >about the syntonic comma problem more than anyone --
>
> Yes, I know you have. I was being bad on account of the last time I
> wrote on this theme, you came back and said, don't worry, [whomever]
> isn't going to hit this problem.
>
> [Paul:]
> >but you have to remember that anyone composing in strict JI has
already
> >had to deal with this problem, or avoid it completely, somehow . .
. so
> >if the decision is between 72-tET and strict JI, this does not
become a
> >deciding factor.
>
> Quite true. My complaint is that 72-tET does not address this
serious
> problem of strict JI. The solution is at hand, but 72-tET
sidesteps it!
>
> JdL

[I wrote:]
>> . It is not anywhere near "excessively close to JI". Its errors
>> through the 11-limit are acceptable to my ear, much better than
>> 12-tET, but 4 cents is far from trivial to many ears.

[Gene wrote:]
>Agreed.

[JdL:]
>> . More importantly, IMHO, is the "straight jacket" it imposes with
>> regard to the syntonic comma. Any music with a normal diatonic
>> progression C-A-D-G-C has either got to drift or shift a comma
>> all at once (howbeit a reduced comma at 16.67 cents).

[Gene:]
>Joseph wanted something close to JI; if he asked for a good meantone
>system the answer would have been different. It's simply a feature,
>not a bug, of JI that your "normal diatonic progression" transposes
>down a comma--5/3 * 2/3 * 4/3 * 2/3 = 80/81 is a fact of arithmetic.

Yes, I understand the arithmetic. What I don't understand is the use
of a tuning system that does not overcome, easily and smoothly, the
challenges of the arithmetic.

[JdL:]
>>Still, I would prefer that a Reinhard type notation (cents deviation
>>from 12-tET) be adopted in place of 72-tET, because it allows commas
>>to be taken on the fly, a few cents at a time, instead of suddenly,
>>painfully.

[Gene:]
>The easy way to do that is meantone, of course.

If one accepts the narrow fifths, which I have trouble with, and if one
accepts the limitations on modulation, which much of the music I'm
interested in does not. Meantone does of course eat the syntonic comma,
but there is a great deal that is possible that meantone does not
support.

I'm biased toward a form of notation that would express the adaptive
qualities that my own work does. To my ear, anything less doesn't cut
the mustard.

JdL

[Bob wrote:]
>Unless we're recreating (i.e., retuning) historical music, then why
>regard this as a "problem" of JI rather than as a PROPERTY INDIGENOUS
>TO IT AND TO BE COMPOSITIONALLY EXPLOITED???????

Sorry, but I have yet to hear a work that brings such a thought from
fantasy to reality. I like being able to deal with the syntonic comma
in small increments. Kind of like your choir does naturally, Bob.

JdL

Yes, 53 was the first microtonal temperament I learned about decades
ago and for a long time thought it was musical Nirvana, although I
never had the option of working in it. Your points are well-taken,
but as I thought I'd made clear, I had specific criteria for this
search, which does not eliminate others or myself from using other
options.

I wanted the least number of equal steps per octave that gave a close
approximation of JI up through 19-limit. The only really rough spot
in 72 is the 13th harmonic, which is 7.2 cents out, but that's half
of what major 3rd is in 12-tET and it's way up the harmonic series,
so not quite so critical in my view.

By the way, has anyone notied that 19-tET is 7 (diatonic) +12, 31 is
12 +19, and 50 (Woolhouse) is 31 +19? The interesting thing is these
are all meantone-equivalent ETs. If you go beyond that, you get
things like 31 + 41 is 72, etc. Curious, huh? Comments, Gene?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > This having been said, the difference between 72-tET and JI is
for
> me
> > musically small enough (not mathematically, Gene) to become
> > practically insignificant, and the above-cited experience would
> > indicate that the difference would be even less significant for
the
> > everwhelming majority of musically-involved humanity.
>
> I would say that for many people you will have reached that point
> already with the 53-et, and for many others the 31-et will sound
> better to them, JI or not. What is significant depends on what you
> are counting as significant; of course with enough steps the
question
> does finally become moot.
>
> > I was looking for the lowest possble number of equal steps per
> octave
> > that would serve as a convenient but still satisfactory
conceptual
> > model for JI. Paul Erlich suggested 72-tET.
>
> And I would suggest the problem is not well-defined, and that you
> could equally well argue that 53 or 171 is the correct answer.
>
> > In addition to many other useful properties, it offers BY FAR the
> > closest rational approximation to the generator we have dubbed
the
> > secor (116.7156 cents), after its author, of any ET until you are
> way
> > up into the hundreds and the step sizes themselves become so
small
> > that the exercise of optimization begins to become meaningless.
>
> And 53 does excellently well for the Orwell (31+22) and Schismatic
> (41+12) generators, and so forth. Trying to decide which of these
> generators, or which et, is "the best" is a but like trying to
decide
> if the "best" place to take a vacation is Italy, Tahiti or Orlando,
> Florida; the various systems are simply structured quite
differently
> and do different things.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Bob wrote:]
> >Unless we're recreating (i.e., retuning) historical music, then
why
> >regard this as a "problem" of JI rather than as a PROPERTY
INDIGENOUS
> >TO IT AND TO BE COMPOSITIONALLY EXPLOITED???????
>
John had said:
> Sorry, but I have yet to hear a work that brings such a thought from
> fantasy to reality.

Bob:
Agreed, but is that not just territory that remains to be explored?
I'm interested in it. The question that remains is my compositional
competence to explore it in a musically successful way (by my own
criteria, lest I get a flood of responses dickering with the
expression "in a musically successful way").

John had said:
I like being able to deal with the syntonic comma
> in small increments. Kind of like your choir does naturally, Bob.
>
Bob:
Yep, but again, we're back to historical music. I'm with you totally
in that context, John. I love the renaiisance choral music just about
above anything, and it's all adaptive JI when it's a cappella, or at
least should be in my humble opinion. I regard the meantone
temperaments of the period as just a kind of frozen adaptive JI for
lack of superior physical alternatives.

--- In tuning@y..., BobWendell@t... wrote:

/tuning/topicId_29265.html#29381

hear more than they
> usually can even though my own experience indicates that I hear
> intonationally better than 99+% of the musical world. I tend,
rather
> than simply accepting that, to project my own ability to hear on
> everyone else. That is typical of us humans. We tend to tacitly,
> unconsciously assume that others' experience parallels our own.
>
> This having been said, the difference between 72-tET and JI is for
me
> musically small enough (not mathematically, Gene) to become
> practically insignificant, and the above-cited experience would
> indicate that the difference would be even less significant for the
> everwhelming majority of musically-involved humanity.
>
> I was looking for the lowest possble number of equal steps per
octave
> that would serve as a convenient but still satisfactory conceptual
> model for JI. Paul Erlich suggested 72-tET. I have looked into its
> properties thoroughly and feel it is quite satisfactory.
>
> In addition to many other useful properties, it offers BY FAR the
> closest rational approximation to the generator we have dubbed the
> secor (116.7156 cents), after its author, of any ET until you are
way
> up into the hundreds and the step sizes themselves become so small
> that the exercise of optimization begins to become meaningless.
>

Wow... Why is it that all the *practical* musicians on this list keep
coming up with 72! I have to say that Paul, Dan Stearns and Joe
Monzo sold me on it practically right away.

I assume you know the notational symbols for it, Bob?

There is an illustration of them in the files section of this very
group:

/tuning/files/Pehrson/Sims.GIF

I happen to believe now that 72-tET is the *only* near JI system that
an *average* performer can learn readily.

I'm really thinking I will use this system for the rest of my life.

Well, actually, I've used it for most of my life already, since I've
used 12-tET... :)

_________ ________ _______
Joseph Pehrson

[I wrote:]
>>Sorry, but I have yet to hear a work that brings such a thought from
>>fantasy to reality.

[Bob:]
>Agreed, but is that not just territory that remains to be explored?
>I'm interested in it. The question that remains is my compositional
>competence to explore it in a musically successful way (by my own
>criteria, lest I get a flood of responses dickering with the
>expression "in a musically successful way").

More power to you, Bob. So far, I've heard lots of speculation about
the potential, but little in the way of music I'd come back to. May
you prove that it's possible!

[JdL:]
>>I like being able to deal with the syntonic comma in small increments.
>>Kind of like your choir does naturally, Bob.

[Bob:]
>Yep, but again, we're back to historical music. I'm with you totally
>in that context, John. I love the renaiisance choral music just about
>above anything, and it's all adaptive JI when it's a cappella, or at
>least should be in my humble opinion. I regard the meantone
>temperaments of the period as just a kind of frozen adaptive JI for
>lack of superior physical alternatives.

I'm all for new music, but my guess is that this trick from the past
(making the syntonic comma vanish) will be useful in a wide range of
music yet to be composed. I do not subscribe to the notion that
methods of this sort must be discarded in order to compose music that
is "not historical".

JdL

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

/tuning/topicId_29265.html#29392

>> Hi John,
>
> Well, I've probably discussed and warned and whined and complained
> about the syntonic comma problem more than anyone -- but you have
to remember that anyone composing in strict JI has already had to
deal
> with this problem, or avoid it completely, somehow . . . so if the
> decision is between 72-tET and strict JI, this does not become a
> deciding factor.

The thing is... if John could somehow design his computer program so
that everybody with a sequencer and MIDI keyboard could use it in
*real time* then, I have to admit, he'd have a real advantage over 72-
tET for anybody wanting to do JI...

However, John's thing is a big "crunching" affair (no offense meant,
just the way it is) and certainly "leisure" rather than "real time"
tuning... so given these constraints it looks like "living with 72"
is a viable alternative!!!

________ _______ ______
Joseph Pehrson

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [I wrote:]
> >>Sorry, but I have yet to hear a work that brings such a thought
from
> >>fantasy to reality.
>
> [Bob:]
> >Agreed, but is that not just territory that remains to be
explored?
> >I'm interested in it. The question that remains is my
compositional
> >competence to explore it in a musically successful way (by my own
> >criteria, lest I get a flood of responses dickering with the
> >expression "in a musically successful way").
>
> More power to you, Bob. So far, I've heard lots of speculation
about
> the potential, but little in the way of music I'd come back to. May
> you prove that it's possible!
>
> [JdL:]
> >>I like being able to deal with the syntonic comma in small
increments.
> >>Kind of like your choir does naturally, Bob.
>
> [Bob:]
> >Yep, but again, we're back to historical music. I'm with you
totally
> >in that context, John. I love the renaiisance choral music just
about
> >above anything, and it's all adaptive JI when it's a cappella, or
at
> >least should be in my humble opinion. I regard the meantone
> >temperaments of the period as just a kind of frozen adaptive JI
for
> >lack of superior physical alternatives.
>
> I'm all for new music, but my guess is that this trick from the
past
> (making the syntonic comma vanish) will be useful in a wide range
of
> music yet to be composed. I do not subscribe to the notion that
> methods of this sort must be discarded in order to compose music
that
> is "not historical".
>
> JdL

Bob:
Agreed! Nope, not necessary. Just another possibility.

--- In tuning@y..., BobWendell@t... wrote:

> I wanted the least number of equal steps per octave that gave a
close
> approximation of JI up through 19-limit.

I didn't know that, but this still is not well-defined without a
notion of "close" to help us out. In any case the 72-division hardly
leaps off the page when it comes to the 19-limit--what about the 50,
80 or 94 ets as competition? The Woolhouse 50-et requires less notes
than 72, it's a meantone system, and it does quite a decent job in
the 19-limit. On the other hand if you have a truly take-no-prisoners
attitude about higher prime limits, the 311-et might be good enough
even for Jacky.

> By the way, has anyone notied that 19-tET is 7 (diatonic) +12, 31
is
> 12 +19, and 50 (Woolhouse) is 31 +19? The interesting thing is
these
> are all meantone-equivalent ETs. If you go beyond that, you get
> things like 31 + 41 is 72, etc. Curious, huh? Comments, Gene?

That's been implicit in some of my recent posts here, including the
one you followed up to, which may be why you are bringing this up, I
suppose. The diatonic scale in various ets I would notate as 7;7+5
(=12), 7;12+7 (=19), 7;19+12 (=31), 7;31+19 (=50). If you keep this
up, you start to get meantone generators as well as other ways of
approximating a fifth which are more accurate. The miracle generator
in the 72-et is 41+31, and I would call Blackjack 21;41+31, Canasta
31;41+31 and so forth. The reason for it is that relative error is
linear.

--- In tuning@y..., jpehrson@r... wrote:

> I happen to believe now that 72-tET is the *only* near JI system
that
> an *average* performer can learn readily.

(1) If the average performer can handle 72, they can also most likely
handle 84.

(2) I think it very unlikely that the average performer will find 72
easier to perform than 19 or 31. The divisibility by 12 is of less
significance than whether you are in a meantone system or not.

> I'm really thinking I will use this system for the rest of my life.

Why not? It's an excellent system. It isn't the *only* system,
however, and it really would make just as much sense to become
religious over 31 or 53--or 50, as I was just suggesting.

> Well, actually, I've used it for most of my life already, since
I've
> used 12-tET... :)

In no way does that count, sorry. :)

[Joseph Pehrson wrote:]
>The thing is... if John could somehow design his computer program so
>that everybody with a sequencer and MIDI keyboard could use it in
>*real time* then, I have to admit, he'd have a real advantage over 72-
>tET for anybody wanting to do JI...

>However, John's thing is a big "crunching" affair (no offense meant,
>just the way it is) and certainly "leisure" rather than "real time"
>tuning... so given these constraints it looks like "living with 72"
>is a viable alternative!!!

Heck, I'm not gonna take offense at the idea that my program does a lot
of "crunching". That's allegedly its power, that it takes a lot of
factors into account and weighs them against each other to try to find
the best tuning compromise at each moment in time.

I don't mean to discourage you from using 72-tET, Joe. I _do_ like to
sound certain notes of caution, as you see. I mainly pipe up when the
issue is expressed in terms of "72-tET: the perfect answer to tuning of
the future?" To this question, I would answer an emphatic "No!" But
for specific works, and even for a composer's lifetime output of works,
it may be the perfect solution. Just be sure to understand what it can
and cannot do.

I consider the "Holy grail", which would combine the power of leisure
retuning and the excitement of live performance, to be an application,
as yet unwritten, that Robert Walker and I (and others?) have talked
about: the performer plays the piece into a MIDI file, tunes that
recording (manually, or using a program such as mine, with additional
manual tweaks if desired), then during the live performance, a computer
cleverly synchronizes the tuning already decided with the precise timing
and velocity (loudness) of each note as it is played. (Did I make it
clear I'm thinking of a piano here? Acoustic horns, winds, and strings
can be tweaked by ear, of course)

And as long as I'm dreaming, this will happen on an acoustic grand piano
which is somehow capable of being tuned on the fly. But I'd settle for
a good synth module and an 88-key weighted keyboard in the short run.
;-)

JdL

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > I wanted the least number of equal steps per octave that gave a
> close
> > approximation of JI up through 19-limit.
>
> I didn't know that, but this still is not well-defined without a
> notion of "close" to help us out.

Bob had a rather clear idea of the notion of "close" he thought would
be relevant for his own music, as he put forward some time ago.

> In any case the 72-division hardly
> leaps off the page when it comes to the 19-limit--what about the
50,
> 80 or 94 ets as competition?

Bob was implying (at the time of the original discussion) putting
more weight (i.e., allowing smaller errors) on the intervals within
the 7-limit or 11-limit (I forget which), and allowing somewhat
larger errors beyond that . . . with such a weighting, and only going
out through the 17-limit, 72 beats 50, 80, and 94. Bob brought up the
19-limit (correct me if I'm wrong, Bob) for the sole case of the
16:19:24 chord, which 72 represents as well as 12 does; i.e., quite
well.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., jpehrson@r... wrote:
>
> > I happen to believe now that 72-tET is the *only* near JI system
> that
> > an *average* performer can learn readily.
>
> (1) If the average performer can handle 72, they can also most
likely
> handle 84.

You forget that most contemporary performers are already trained in
quartertones. And that, 72 is somewhat better than 84 in the 7-limit
and 9-limit and far better in the 11-limit.
>
> (2) I think it very unlikely that the average performer will find
72
> easier to perform than 19 or 31. The divisibility by 12 is of less
> significance than whether you are in a meantone system or not.

For performance purposes, I have to testify that you are quite
mistaken here, Gene. Today, sadly, one finds absolutely no meantone
tendencies in all but the most specialized of early-music ensembles --
and Bob's choir, of course. Seriously, though, modern classical
performers stick to near-400¢ major thirds, have absolutely no
experience narrowing fifths in order to facilitate purer thirds, and
are out at sea when it comes to 19 or 31. Meanwhile, the subsets of
72-tET that Joseph is working with, such as Blackjack, 18-tET, and
perhaps Canasta in the future, don't have any single-comma steps, so
subversion (or confirmation) of meantone expectations won't be
possible anyway.

> > I'm really thinking I will use this system for the rest of my
life.
>
> Why not? It's an excellent system. It isn't the *only* system,
> however, and it really would make just as much sense to become
> religious over 31 or 53--or 50, as I was just suggesting.

In the 24-tET-oriented world that the instrumental performers Joseph
is likely to be working with are coming from, 72 makes a lot more
sense. Me, I'm going to be working mainly in 22 and 31 for the
forseeable future, as I'm not terribly interested in composing for
players of traditional instruments.

--- In tuning@y..., genewardsmith@j... wrote:

/tuning/topicId_29265.html#29423

> --- In tuning@y..., jpehrson@r... wrote:
>
> > I happen to believe now that 72-tET is the *only* near JI system
> that an *average* performer can learn readily.
>
> (1) If the average performer can handle 72, they can also most
likely handle 84.
>
> (2) I think it very unlikely that the average performer will find
72 easier to perform than 19 or 31. The divisibility by 12 is of
less significance than whether you are in a meantone system or not.
>

Hello Gene!

Man, I wish I could understand more of your posts... they look very
cool, but the sponge is only picking up about half of them. Maybe on
the "re-reads..."

True, I was also considering 19 and 31 but the fact that the basic
note names change from their "accustomed" frequency is a problem for
me.

For example, 19 is very easy to notate, of course, since it is a
basic meantone, if notated as non-enharmonic pitches. HOWEVER, the
problem I have is that the basic "note names" are not the same
frequency as 12-tET. Maybe *some* people get used to that... Neil
Haverstick obviously has.

However, for other "general" performers I could see a problem with
it... I would much rather start with a system that has a basis that
is *exactly* the same as the 12-tET that everybody has practiced and
studied... From that angle, it seems "easier" to me...

> I'm really thinking I will use this system for the rest of my life.
>
> Why not? It's an excellent system. It isn't the *only* system,
> however, and it really would make just as much sense to become
> religious over 31 or 53--or 50, as I was just suggesting.
>

Religion is a big problem right now, so let's say "enthusiasm"
instead... :)

> > Well, actually, I've used it for most of my life already, since
> I've used 12-tET... :)
>
> In no way does that count, sorry. :)

What if I were to write a piece solely in 12-tET and then added a
final note that was 1/6th of a whole tone??

Maybe that's a new "innovative" direction for me.... :)

_________ _______ _______
Joseph Pehrson

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:
> > --- In tuning@y..., BobWendell@t... wrote:
> >
> > > I wanted the least number of equal steps per octave that gave a
> > close
> > > approximation of JI up through 19-limit.
> >
Gene:
> > I didn't know that, but this still is not well-defined without a
> > notion of "close" to help us out.
>
Paul:
> Bob had a rather clear idea of the notion of "close" he thought
would
> be relevant for his own music, as he put forward some time ago.
>
Bob answers now:
Yes, I had specified an ET with minimum steps per octave that was
within 5 cents of JI intervals through 19-limit. 72 has only one
prime (13 is -7.2 cents off) that violates this. If I were going to
pick a prime that was least important for me musically in terms of
accuracy, 13 would have been it. Besides that's still half the error
of the very important prime 5 in 12-tET!!!

And you have to go to such high numbers to improve this situation
that the interval sizes themselves start to approach the error limits
I specified, so you havent' really bought anything with THAT!

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> I consider the "Holy grail", which would combine the power of
leisure
> retuning and the excitement of live performance, to be an
application,
> as yet unwritten, that Robert Walker and I (and others?) have
talked
> about: the performer plays the piece into a MIDI file, tunes that
> recording (manually, or using a program such as mine, with
additional
> manual tweaks if desired), then during the live performance, a
computer
> cleverly synchronizes the tuning already decided with the precise
timing
> and velocity (loudness) of each note as it is played. (Did I make
it
> clear I'm thinking of a piano here? Acoustic horns, winds, and
strings
> can be tweaked by ear, of course)
>
> And as long as I'm dreaming, this will happen on an acoustic grand
piano
> which is somehow capable of being tuned on the fly.

Somehow I don't think all this would be too helpful to Joseph
Pehrson, who is interested in writing music that is TRULY MICROTONAL.

[Paul wrote:]
>Somehow I don't think all this would be too helpful to Joseph
>Pehrson, who is interested in writing music that is TRULY MICROTONAL.

What the heck is _that_ supposed to mean?

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Paul wrote:]
> >Somehow I don't think all this would be too helpful to Joseph
> >Pehrson, who is interested in writing music that is TRULY
MICROTONAL.
>
> What the heck is _that_ supposed to mean?
>
> JdL

Meaning, it's not usually based on a 12 tones per octave "input
file", if you catch my drift, and it's often based on exploiting
dissonances for their own sake, rather than attempting to maximize
the consonance of every sonority -- witness Joseph's use of 18-tET,
which I'm sure he'd rather remain fixed than be adaptively tweaked to
give nice consonances. . . .of course I'd rather Joseph speak for
himself . . .

[Paul wrote:]
>>>Somehow I don't think all this would be too helpful to Joseph
>>>Pehrson, who is interested in writing music that is TRULY MICROTONAL.

[I wrote:]
>>What the heck is _that_ supposed to mean?

[Paul:]
>Meaning, it's not usually based on a 12 tones per octave "input
>file", if you catch my drift, and it's often based on exploiting
>dissonances for their own sake, rather than attempting to maximize
>the consonance of every sonority -- witness Joseph's use of 18-tET,
>which I'm sure he'd rather remain fixed than be adaptively tweaked to
>give nice consonances. . . .of course I'd rather Joseph speak for
>himself . . .

The techniques I speak of could be applied to any music. They would be
unnecessary if one were satisfied with a fixed number of pitches per
octave, but might be useful in any other circumstances, when the actual
playing is via keys. The technique is not tied to a number of tones
per octave or to consonance vs. "dissonances for their own sake".

JdL

Only a periodic wave that starts at time minus infinity and goes to
infinity has a precise fundamental frequency. One that starts and
stops has an uncertainty in the frequency inversely proportional to
the duration. (The constant of proportionality depends on the envelope
and the statistical measure of deviation being used).

The Fourier transform takes as a fundamental assumption that the
sample you give it is (and was) repeated endlessly. If it was not,
then _you_ must take the classical uncertainty principle into
account because _it_ will not.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> Only a periodic wave that starts at time minus infinity and goes to
> infinity has a precise fundamental frequency. One that starts and
> stops has an uncertainty in the frequency inversely proportional to
> the duration. (The constant of proportionality depends on the
envelope
> and the statistical measure of deviation being used).

Did you see Bob Wendell's rebuttal to this assertion (which I made
myself at the start of this thread)? He said, that in the classical
case, what you are calling "uncertainty" is not really uncertainty at
all but merely bandwidth. He made the point that, if the tone starts
and stops, the _center_ of this bandwidth will give you a precise
measure of the frequency. This is in contrast with the quantum case,
where one cannot scan the bandwidth to find the center, since a
single measurement immediately "collapses" the distribution, and the
bandwidth is indeed interpretable as an uncertainty. Anyway, when I
asked him, what if rather than being preceded and followed by
silence, it's preceded and followed by another tone, Bob became
evasive, claiming this was not in the scope of the original
discussion. I disagree.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > Only a periodic wave that starts at time minus infinity and goes
to
> > infinity has a precise fundamental frequency. One that starts and
> > stops has an uncertainty in the frequency inversely proportional
to
> > the duration. (The constant of proportionality depends on the
> envelope
> > and the statistical measure of deviation being used).
>
> Did you see Bob Wendell's rebuttal to this assertion (which I made
> myself at the start of this thread)? He said, that in the classical
> case, what you are calling "uncertainty" is not really uncertainty
at
> all but merely bandwidth. He made the point that, if the tone starts
> and stops, the _center_ of this bandwidth will give you a precise
> measure of the frequency. This is in contrast with the quantum case,
> where one cannot scan the bandwidth to find the center, since a
> single measurement immediately "collapses" the distribution, and the
> bandwidth is indeed interpretable as an uncertainty.

Ah! The point missing here is that (as far as I am aware) our ears
don't hear bandwidths, only pitches.

> Anyway, when I
> asked him, what if rather than being preceded and followed by
> silence, it's preceded and followed by another tone, Bob became
> evasive, claiming this was not in the scope of the original
> discussion. I disagree.

I agree with you Paul. Bob. I think you're getting carried away by the
maths and forgetting that you lose information from the Laplace
transform (or whatever continuous spectrum transform you're using)
when you go to the ear-brain system.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > > Only a periodic wave that starts at time minus infinity and
goes
> to
> > > infinity has a precise fundamental frequency. One that starts
and
> > > stops has an uncertainty in the frequency inversely
proportional
> to
> > > the duration. (The constant of proportionality depends on the
> > envelope
> > > and the statistical measure of deviation being used).
> >
> > Did you see Bob Wendell's rebuttal to this assertion (which I
made
> > myself at the start of this thread)? He said, that in the
classical
> > case, what you are calling "uncertainty" is not really
uncertainty
> at
> > all but merely bandwidth. He made the point that, if the tone
starts
> > and stops, the _center_ of this bandwidth will give you a precise
> > measure of the frequency. This is in contrast with the quantum
case,
> > where one cannot scan the bandwidth to find the center, since a
> > single measurement immediately "collapses" the distribution, and
the
> > bandwidth is indeed interpretable as an uncertainty.
>
> Ah! The point missing here is that (as far as I am aware) our ears
> don't hear bandwidths, only pitches.

Well, I've read that there is evidence that the data from the
cochlear membrane is used again later in processing, even after the
rough assignment from pitch to place has taken place . . . one
shouldn't rule out a more sophisticated mechanism going on in the ear-
brain system.
>
> > Anyway, when I
> > asked him, what if rather than being preceded and followed by
> > silence, it's preceded and followed by another tone, Bob became
> > evasive, claiming this was not in the scope of the original
> > discussion. I disagree.
>
> I agree with you Paul. Bob. I think you're getting carried away by
the
> maths and forgetting that you lose information from the Laplace
> transform (or whatever continuous spectrum transform you're using)
> when you go to the ear-brain system.

I think Robert Walker was getting even more carried away by the math.
There is no doubt that information is lost. The question is how much,
and how much of it higher-order processing in the brain can recover.

Hi Dave,

That's a nice clear expo. Helps explain why with the FFT
the bin spacing depends only on the length of the clip and
not the sampling rate.

Do you know why peak interpolation works then?

It certainly does - you do a FFT for say a one second sample
with peak interpolation and get the same result and similar accuracy
as you get with perhaps a 30 second sample without.

I'm approaching this as a practical matter and haven't gone into
the theory at all deeply.

For the peak interpolation I'm using standard methods for
interpolating FFT peaks, not ones that I invented myself.

I know about windowing, and have options for that in FTS.
In practice the various FFT windows seem to make surprisingly little
difference to the result. Just use a rectangular window, i.e.
sharp cut off, and you still get good values for the frequencies
of the partials with peak interpolation.

Robert

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

/tuning/topicId_29265.html#29451

>
> Meaning, it's not usually based on a 12 tones per octave "input
> file", if you catch my drift, and it's often based on exploiting
> dissonances for their own sake, rather than attempting to maximize
> the consonance of every sonority -- witness Joseph's use of 18-tET,
> which I'm sure he'd rather remain fixed than be adaptively tweaked
to
> give nice consonances. . . .of course I'd rather Joseph speak for
> himself . . .

It seems pretty clear to me that John's activities work best in the
realm of "experimental retuning" of pieces from the standard or pop
repertoire that utilize 12 tones per octave. To my knowledge, John
hasn't developed other bases than 12, so my assessment can be
quantifiably verified... :)

Let me state right up front that I am a *great* admirer of John's
efforts. In fact, when I first started on this list I was totally
mesmerized by them. I think I enjoyed looking at all the columns of
numbers. You know, I really like that kind of thing, even if I don't
always understand it... :)

In any case, *then* I heard the music and I was *totally bowled
over...* I just *loved* the various "warped" classics which John was
claiming were superior and other "history minded" musicians and music
aficionados thought were the most deplorable things on the planet.

I was all for them, though. In fact, I am, generally speaking,
rather tired of *much* of the standard repertoire, particularly the
*Romantic* period, and any way these pieces can be "twisted"
or "distorted" really picks up my interest in them!

Actually, I lie. That Schubert Sonata that John chose to retune
[ummm, G major Op. 78... I just checked my score] meant as much to me
as anybody. It was thrilling... and I'm serious about that. I guess
I just have to be in the right mood for the Romantics... and John
really put me there with his various retunings.

I *loved* to compare his version with the 12-tET versions and
consider my preference...

By the way, what *were* the results of John's various listening
experiments where listeners were asked their preferences?? Were
there not enough listeners to conclude that study??

My own personal preferance has almost *always* been for John's just
tunings... But mostly due to the fact that I really am truly tired
and sick of 12-tET, wonderful as it is and has been (forever...)

On the other hand, for *me* anyway, don't mess with the Medieval and
Renaissance repertoire. There I want as "authentic" an experience as
historians can give us... and of course with Margo Schulter right
here, we have an incredible "in house" expert...

However, Medieval and Renaissance music doesn't seem "hackneyed" like
the Romantics. That music is fresh because we look at it through the
long prism of time and, frankly, are still discovering it... (At
least *I* am...! :)

_________ _______ ________
Joseph Pehrson

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

> > (2) I think it very unlikely that the average performer will find
> 72
> > easier to perform than 19 or 31. The divisibility by 12 is of
less
> > significance than whether you are in a meantone system or not.

> For performance purposes, I have to testify that you are quite
> mistaken here, Gene. Today, sadly, one finds absolutely no meantone
> tendencies in all but the most specialized of early-music
ensembles --
> and Bob's choir, of course.

I think it is a little broader than that, since "early music" now
means the high classical period of Mozart, Haydn and Beethoven, with
some degree of incursion into the early romantic era. In any case,
meantone is what people did when they composed a lot of CP music, and
most certainly not, ever, 72-et. Given that fact, it's almost bound
to be easier. The notation, standard sharps and flats, seems to win
in the ease category also.

The bottom line is that people are, right now, performing quite a bit
in meantone, and are doing so increasingly. That means far more
performance practice than 72-et is getting, so I don't think you've
got much of a factual basis for this claim.

Seriously, though, modern classical
> performers stick to near-400¢ major thirds, have absolutely no
> experience narrowing fifths in order to facilitate purer thirds,
and
> are out at sea when it comes to 19 or 31.

"Modern classical performers" include those who perform meantone as a
matter of course these days--it isn't all the BPO anymore.

> > Why not? It's an excellent system. It isn't the *only* system,
> > however, and it really would make just as much sense to become
> > religious over 31 or 53--or 50, as I was just suggesting.

> In the 24-tET-oriented world that the instrumental performers
Joseph
> is likely to be working with are coming from, 72 makes a lot more
> sense.

I still think the transition to 19, 31 or even 50 would be easier,
but I don't look at it as much of a problem. If you are going to
learn a system, then learn it. Your 10 or 12 notes out of 22 makes as
much sense as anything from that point of view.

Me, I'm going to be working mainly in 22 and 31 for the
> forseeable future, as I'm not terribly interested in composing for
> players of traditional instruments.

I think 22 and 31 are perhaps the two best tries out there for
someone wanting to do instrumental music with full transpostional
possibilities. Go for it, sez I. As for me, I'm sticking with
electronic; that way I do what I want, as I want. Besides, I'm too
clumsy to be an instrumentalist!

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > Only a periodic wave that starts at time minus infinity and goes
to
> > infinity has a precise fundamental frequency. One that starts and
> > stops has an uncertainty in the frequency inversely proportional
to
> > the duration. (The constant of proportionality depends on the
> envelope
> > and the statistical measure of deviation being used).
>
> Did you see Bob Wendell's rebuttal to this assertion (which I made
> myself at the start of this thread)? He said, that in the classical
> case, what you are calling "uncertainty" is not really uncertainty
at
> all but merely bandwidth. He made the point that, if the tone
starts
> and stops, the _center_ of this bandwidth will give you a precise
> measure of the frequency. This is in contrast with the quantum
case,
> where one cannot scan the bandwidth to find the center, since a
> single measurement immediately "collapses" the distribution, and
the
> bandwidth is indeed interpretable as an uncertainty. Anyway, when I
> asked him, what if rather than being preceded and followed by
> silence, it's preceded and followed by another tone, Bob became
> evasive, claiming this was not in the scope of the original
> discussion. I disagree.

Bob answers:
Sorry, Paul! I actually had not understood your elaboration of this
question until your last previous post, I believe it was, in this
thread.

In this spin-off question the issue becomes: "What happens if we do
not interrupt the frequency, but change it suddenly to another
frequency and then to a third (perhaps the original frequency
again)?"
This is an FM or frequency modulation where the modulating signal is
a radically low-frequency step-wave (in a scale, or square wave in
the case of a return to the original pitch).

In music the stepped rate of change in frequency is of an extremely
low frequency (a few cycles per second at most), yet the magnitude of
the frequency change represents a very large percentage of the
original frequency. This is quite the opposite from the kind of
modulation used in FM radio, where the frequency variations of the
carrier are vanishingly small compared to the absolute frequency
being modulated and the modulating frequencies are very high compared
to the very low rate of stepped changes under consideration here.

Although I'm not at all familiar with the mathematical rigors of this
kind of unusual "FM", in such an extreme case it should be very safe
to say that with a step frequency that is extremely low even at fast
tempos and short note values (say 32nd or 64th notes at a vivace
tempo) vis-a-vis the frequency corresponding to pitch perception, the
bandwidth introduced by these sudden stepped frequency changes would
have next to zero effect on the pitch perceived. What would become
significant much earlier as we increase tempo would be the
limitations of the human perceptual apparatus in detecting pitches of
extremely short duration.

In short, in my carefully considered opionion, this whole question of
classical uncertainty with regard to pitch of finite duration is more
academic than practical. The indeterminacy it might theoretically
impose in principle on the accuracy of pitch perception is
vanishingly small as compared to the limitations, neurological and
otherwise, of the mechanics of human aural perception.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > (2) I think it very unlikely that the average performer will
find
> > 72
> > > easier to perform than 19 or 31. The divisibility by 12 is of
> less
> > > significance than whether you are in a meantone system or not.
>
> > For performance purposes, I have to testify that you are quite
> > mistaken here, Gene. Today, sadly, one finds absolutely no
meantone
> > tendencies in all but the most specialized of early-music
> ensembles --
> > and Bob's choir, of course.
>
> I think it is a little broader than that, since "early music" now
> means the high classical period of Mozart, Haydn and Beethoven,

Beethoven makes very heavy use of enharmonic equivalence -- meantone
is pretty much out the window.

with
> some degree of incursion into the early romantic era. In any case,
> meantone is what people did when they composed a lot of CP music,
and
> most certainly not, ever, 72-et. Given that fact, it's almost bound
> to be easier.

As a mathematician, one would certainly think so. As a musician
living in the world today, it unfortunately isn't the case.
Conservatories train all instrumentalists in 12-tET, and training
overwhelms musical logic.

> The notation, standard sharps and flats, seems to win
> in the ease category also.

That's again because of the common practice having begun with
meantone.
>
> The bottom line is that people are, right now, performing quite a
bit
> in meantone, and are doing so increasingly.

Well, based on months and months of discussions before you joined us,
Gene, I don't think Joseph's met too many of these people, let alone
any who would even be willing to attempt any new, microtonal music --
and Joseph's certainly no hermit in the New York music scene (well,
at least he wasn't before Sept. 11th . . .), if I may speak for
Joseph again.

> That means far more
> performance practice than 72-et is getting, so I don't think you've
> got much of a factual basis for this claim.

The performers who are interested in playing new music are trained in
12-tET (very equal, as appropriate for serialism) and 24-tET, and
most definitely not meantone.

> Seriously, though, modern classical
> > performers stick to near-400¢ major thirds, have absolutely no
> > experience narrowing fifths in order to facilitate purer thirds,
> and
> > are out at sea when it comes to 19 or 31.
>
> "Modern classical performers" include those who perform meantone as
a
> matter of course these days--it isn't all the BPO anymore.

BPO?

> I still think the transition to 19, 31 or even 50 would be easier,

Most instrumentalists coming out of the conservatories have spent
years and years honing their ability to produce precise 12-tET, and
in some cases 24-tET, pitches on their instruments. With 19, 31, or
50, they would have to learn 9, 15, or 25 new types of alterations to
the pitches they've been trained to produce on demand. With 72, only
3, or 1, new types of alterations are needed.

> I think 22 and 31 are perhaps the two best tries out there for
> someone wanting to do instrumental music with full transpostional
> possibilities. Go for it, sez I.

Perhaps Fortuitously, Harold Fortuin just (on Friday) presented his
Clavette and Microzone generalized keyboards at New England
Conservatory, with most of the music in 22 or 31 -- but also some in
27.35! He said he was looking for musicians to collaborate with and
ultimately present a concert with -- of course I jumped at the
opportunity!

--- In tuning@y..., jpehrson@r... wrote:

> For example, 19 is very easy to notate, of course, since it is a
> basic meantone, if notated as non-enharmonic pitches. HOWEVER, the
> problem I have is that the basic "note names" are not the same
> frequency as 12-tET. Maybe *some* people get used to that... Neil
> Haverstick obviously has.

I'm a really bad violinist, but at least my intonation is good, so
I'll venture to suggest that tuning the strings by the right flat
fifths and practicing scales should work for them, at least. This is,
after all, what people used to do! A keyboard instrument tuned as you
would like it would no doubt help. However it's being done, the fact
is it *is* being done, and there's less of a step from a chamber
orchesta playing in meantone intonation to the 19 or 31 et than there
would be in trying to get to 72-et.

I'm waiting for some brave soul to get an orchesta going with strings
tuned to sharp fifths and playing in 22-et, myself.

--- In tuning@y..., BobWendell@t... wrote:

> Although I'm not at all familiar with the mathematical rigors of
this
> kind of unusual "FM", in such an extreme case it should be very
safe
> to say that with a step frequency that is extremely low even at
fast
> tempos and short note values (say 32nd or 64th notes at a vivace
> tempo) vis-a-vis the frequency corresponding to pitch perception,
the
> bandwidth introduced by these sudden stepped frequency changes
would
> have next to zero effect on the pitch perceived.

Doesn't the Classical Uncertainty Principle apply in full force here?
And does your "find the center" trick still work in this case?

> What would become
> significant much earlier as we increase tempo would be the
> limitations of the human perceptual apparatus in detecting pitches
of
> extremely short duration.

You may be underestimating the degree to which these limitations are
related to the Classical UP -- consider what happens to the cochlea
when presented with these sorts of impulses.

> In short, in my carefully considered opionion, this whole question
of
> classical uncertainty with regard to pitch of finite duration is
more
> academic than practical. The indeterminacy it might theoretically
> impose in principle on the accuracy of pitch perception is
> vanishingly small as compared to the limitations, neurological and
> otherwise, of the mechanics of human aural perception.

Vanishingly small? I thought you had been saying the theoretical
indeterminacy implied by the Classical UP was, at least in some
cases, _larger_ than the actual uncertainty involved in human aural
perception.

--- In tuning@y..., genewardsmith@j... wrote:
>
> I'm waiting for some brave soul to get an orchesta going with
strings
> tuned to sharp fifths and playing in 22-et, myself.

I should make this happen within my lifetime . . . starting to hang
out at NEC more . . . maybe I'll even apply and get a doctorate . . .
of course, a doctorate in a more technical field might be more likely
to pay for itself . . .

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

> > I think it is a little broader than that, since "early music" now
> > means the high classical period of Mozart, Haydn and Beethoven,

> Beethoven makes very heavy use of enharmonic equivalence --
meantone
> is pretty much out the window.

We don't know how Beethoven tuned his piano, and it became moot after
he went deaf, of course, but some form of well-temperament seems
likely.

> Gene, I don't think Joseph's met too many of these people, let
alone
> any who would even be willing to attempt any new, microtonal music -

They are probably much too busy singing Tallis or plinking away on a
fortepiano, but they are certainly out there.

> BPO?

Berlin Philharmonic--you know, *the* orchestra.

> > I still think the transition to 19, 31 or even 50 would be easier,

> Most instrumentalists coming out of the conservatories have spent
> years and years honing their ability to produce precise 12-tET, and
> in some cases 24-tET, pitches on their instruments.

Years working on 24-et? Who'se telling them to do that??

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > I think it is a little broader than that, since "early music"
now
> > > means the high classical period of Mozart, Haydn and Beethoven,
>
> > Beethoven makes very heavy use of enharmonic equivalence --
> meantone
> > is pretty much out the window.
>
> We don't know how Beethoven tuned his piano, and it became moot
after
> he went deaf, of course, but some form of well-temperament seems
> likely.

Yup.
>
> > Gene, I don't think Joseph's met too many of these people, let
> alone
> > any who would even be willing to attempt any new, microtonal
music -
>
> They are probably much too busy singing Tallis or plinking away on
a
> fortepiano, but they are certainly out there.

Let's find them!
>
> > BPO?
>
> Berlin Philharmonic--you know, *the* orchestra.

They can play in meantone? Or they can't? I forget the context.

> Years working on 24-et? Who'se telling them to do that??

24-tET is pretty much _standard_ in the new music world . . . I'll
let Joseph elaborate . . .

--- In tuning@y..., genewardsmith@j... wrote:

/tuning/topicId_29265.html#29471

>
> I'm a really bad violinist, but at least my intonation is good, so
> I'll venture to suggest that tuning the strings by the right flat
> fifths and practicing scales should work for them, at least.

Hi Gene!

Well, it *should* possibly, but most string players have their ears
set to 12-tET... even though some claim otherwise... or at least play
Pythagorean. I believe Paul has some studies to back this up.

Maybe the reason that Neil Haverstick (a very talented musician, by
the way, if you haven't met him yet) is so successful with 19-tET is
the fact that his guitar is FRETTED that way... So he just has to
learn the notes.

At least that would be *my* guess. That's a far different case than
a violinist...

best,

________ _______ _______
Joseph Pehrson

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

> > Berlin Philharmonic--you know, *the* orchestra.

> They can play in meantone? Or they can't? I forget the context.

They are Big Band traditionalists--tuned sharp, very well-trained,
but not at all into authenticity.

> 24-tET is pretty much _standard_ in the new music world . . . I'll
> let Joseph elaborate . . .

When I started looking at this stuff, the first thing I noticed was
22, and the second 31, and they still look good to me. I suppose that
means the "new music world" isn't very mathemically oriented.

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

/tuning/topicId_29265.html#29473

> --- In tuning@y..., genewardsmith@j... wrote:
> >
> > I'm waiting for some brave soul to get an orchesta going with
> strings tuned to sharp fifths and playing in 22-et, myself.
>
> I should make this happen within my lifetime . . . starting to hang
> out at NEC more . . . maybe I'll even apply and get a
doctorate . . . of course, a doctorate in a more technical field
might be more likely to pay for itself . . .

Ummm... it's a nice *honor,* Paul, but before you get started in that
direction, I would urge you to use your mathmatical abilities to find
a high paying "day job" and organize and pay for your *own* orchestra!

Actually, a gentleman by the name of Eric Grunin has done just that
in New York. Well, he lost his job as a computer programmer and had
angry players running after him for a while... but I hear he may get
started with it again. He was, actually, quite successful... big
reviews in the Times, etc., etc..

_________ _______ ______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., genewardsmith@j... wrote:

> Well, it *should* possibly, but most string players have their ears
> set to 12-tET... even though some claim otherwise... or at least
play
> Pythagorean. I believe Paul has some studies to back this up.

String players often tune by ear, by fifths. They would need to have
their hands slapped, be told to go practice their scales, and
*always* to tune flat, using a correctly tuned pitchpipe. Probably
most wouldn't like it, but at least the some of them have learned to
do it.

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

/tuning/topicId_29265.html#29475

> > Years working on 24-et? Who'se telling them to do that??
>
> 24-tET is pretty much _standard_ in the new music world . . . I'll
> let Joseph elaborate . . .

My own experience is that *most* players under about 30 can play and,
frankly, are *expected* to play quarter-tones as part of their
performing arsenal... (more army talk for the G.I.s)

The reason is, most of them encounter at least *ONE* microtonal piece
that uses them by now, and, in learning it, incorporate that
technique into their playing....

At least, that's what I've been hearing from "traditional"
instrumentalists of about that age...

_______ ______ _______
Joseph Pehrson

--- In tuning@y..., genewardsmith@j... wrote:

/tuning/topicId_29265.html#29477

>
> When I started looking at this stuff, the first thing I noticed was
> 22, and the second 31, and they still look good to me. I suppose
that means the "new music world" isn't very mathemically oriented.

Hi Gene!

Well, actually I think a *lot* of musicians enjoy math and some are
quite good at it... take composers Milton Babbitt and Easley
Blackwood, for example....

However, when it comes to composing and performing *tradition* plays
a great role in music and the arts. I believe a *much* greater role
than in the sciences.

When something new is discovered in the sciences, other former,
traditional BS is logically thrown out the window as it should be.

However, in music and the arts, this "logic" is controverted by
tradition... So I suppose cultural tradition plays the same role as
superstition... But it sure is big in the arts!

_________ ________ ______
Joseph Pehrson

--- In tuning@y..., BobWendell@t... wrote:

> Yes, I had specified an ET with minimum steps per octave that was
> within 5 cents of JI intervals through 19-limit. 72 has only one
> prime (13 is -7.2 cents off) that violates this. If I were going to
> pick a prime that was least important for me musically in terms of
> accuracy, 13 would have been it. Besides that's still half the
error
> of the very important prime 5 in 12-tET!!!

The 72-et has a maximum error of 9.68 cents in the 19-limit, so you
are being generous with your initial requirements. If you are a bit
more generous, the 50-et with an error of 12.59 cents might do; if
you are a tad less, you will hold out for 80 (7.25 cents), 94 (5.92
cents) 111 (5.19 cents), or the full monte--121 (4.35 cents.) Nor is
the 72-et the only one to benefit from being less careful with 13--
the 118-et really can claim to be effectively JI in the 5-limit in a
way the 72-et can't get close to; it has an error of 6.8 cents in the
19-limit, but if we ignore 13 it becomes less than half that. Then
there is 171, which has an error of 5.84 in the 19-limit and is
effectively JI up to the 7-limit.

> And you have to go to such high numbers to improve this situation
> that the interval sizes themselves start to approach the error
limits
> I specified, so you havent' really bought anything with THAT!

I don't think 80 is so much larger than 72 that it qualifies as being
a high number while 72 is not, but if that worries you, what about
the attractiveness of 50, which is considerably smaller?

[Joseph Pehrson wrote:]
>It seems pretty clear to me that John's activities work best in the
>realm of "experimental retuning" of pieces from the standard or pop
>repertoire that utilize 12 tones per octave. To my knowledge, John
>hasn't developed other bases than 12, so my assessment can be
>quantifiably verified... :)

Most of my work so far has been with 12 notes per octave. I _do_ have
preliminary code that splits grounding points for selected pitch
classes, yielding up to 24 notes per octave.

On the planning board: interpreting input pitch bends as defining steps.
For example, a piece in 19-tET, realized with pitch bends, would be
understood to represent 19-tET, and adaptive interval adjustments would
be added overtop of grounding, giving better fifths and major thirds
than 19-tET does on its own, while preserving the basic identity of the
19 steps.

I don't believe that my techniques work "best" on music from the past.
Several list members use leisure adaptive tuning on their new works.
However, I'm certainly not going to claim that _all_ music needs to be
run thru my program! I wouldn't have a clue what to do with a 13-tET
piece, for example, other than leave it alone. Ditto, I'm guessing, one
of your Blackjack pieces, Joe.

I'm not quite sure what your subject line means, but I'm glad you enjoy
my treatments. The poll, BTW, has had only 4 responses; it's still out
there if anyone wants to listen. There's more pieces (and polls) up on
crazy_music, and they've also gotten little response so far.

JdL

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

/tuning/topicId_unknown.html#29483

> Paul,
>
> Perhaps if you just say that quartertones (and not 24-tet) are way
old hat by now, and the reason why this is relevant is because this
is an easily doable alteration or embellishment on what's taught and
already firmly under most schooled folks hands and mindset--twelve-
tone equal temperament, then maybe Gene can begin to see why the
learning curve
> of 72-tet (or sixthtones) is an entirely different beast than a
> (seemingly simpler) leap to 19- or 31-tet...
>
> Musicians who can handle quartertones, and that's most every
> conservatory boy and girl who's touching anything resembling
> contemporary classical music, can readily get their heads around
> splitting that into threes--given 72s resources this is not trivial!
>
> --Dan Stearns

Hi Dan!
This sure makes sense to me... You, Paul and Monzo "sold" me on 72-
tET after just a couple of posts... There's no turning back now...
at least for *me*...

_________ _______ ________
Joseph Pehrson

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > Although I'm not at all familiar with the mathematical rigors of
> this
> > kind of unusual "FM", in such an extreme case it should be very
> safe
> > to say that with a step frequency that is extremely low even at
> fast
> > tempos and short note values (say 32nd or 64th notes at a vivace
> > tempo) vis-a-vis the frequency corresponding to pitch perception,
> the
> > bandwidth introduced by these sudden stepped frequency changes
> would
> > have next to zero effect on the pitch perceived.
>
> Doesn't the Classical Uncertainty Principle apply in full force
here?
> And does your "find the center" trick still work in this case?
>
> > What would become
> > significant much earlier as we increase tempo would be the
> > limitations of the human perceptual apparatus in detecting
pitches
> of
> > extremely short duration.
>
> You may be underestimating the degree to which these limitations
are
> related to the Classical UP -- consider what happens to the cochlea
> when presented with these sorts of impulses.
>
> > In short, in my carefully considered opionion, this whole
question
> of
> > classical uncertainty with regard to pitch of finite duration is
> more
> > academic than practical. The indeterminacy it might theoretically
> > impose in principle on the accuracy of pitch perception is
> > vanishingly small as compared to the limitations, neurological
and
> > otherwise, of the mechanics of human aural perception.
>
> Vanishingly small? I thought you had been saying the theoretical
> indeterminacy implied by the Classical UP was, at least in some
> cases, _larger_ than the actual uncertainty involved in human aural
> perception.

Bob:
In the first, fixed frequency case I was saying that the theoretical
uncertainty is not really uncertainty at all. Then it doesn't matter
how large it is compared to the uncertainty intrinsic to the
mechanics of the perceptual apparatus, does it? I meant theoretical
in its final, adjusted sense after taking this into account. Theories
are only as valid as the appropriateness of their application allows
them to be. As I'm sure you're aware, theories are neither true nor
false, but simply represent modeling tools. They are only as good as
the practical value of their explanatory/predictive power.

But you're right. As applied to the second stepped frequency case,
the statement above was not clear. I intended to communicate that
similarly, whatever APPARENT uncertainties might be introduced by
bandwidth considerations would not be confusing to the ear in
determining pitch any more than the other. FM after all, although
more complex, still has symmetrical bandwidth products around the
modulated frequency (or frequencies), regardless of the symmetry or
asymmetry of the modulating signal.

This following argument is admittedly more intuitive than rigorous,
but may be more telling for some readers. I can't see how switching
frequency in what is musically equivalent to a legato style of
articulation would introduce a more confusing bandwidth scenario than
would articulating each frequency separately with a start and stop
for each note. Sometimes good old-fashioned gut intuition is better
than the state of our scientific arts (and don't let anyone convince
you that the term "state of the art" does not apply to science).

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

/tuning/topicId_29265.html#29490

> I'm not quite sure what your subject line means, but I'm glad you
enjoy my treatments.

Just joking around with you, John... Bravo for your achievements!

You are possibly the most successful "adaptive tuner" in history!
Seriously.

However, you have to publish this stuff in some kind of journal. It
wouldn't be, necessarily, a "new music" journal since, I feel, this
subject is of a more "general" nature.

I would write it up and send it out if I were you, so that you'll
be "discovered" one of these days!

_________ _________ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., genewardsmith@j... wrote:
>
> /tuning/topicId_29265.html#29471
>
> >
> > I'm a really bad violinist, but at least my intonation is good,
so
> > I'll venture to suggest that tuning the strings by the right flat
> > fifths and practicing scales should work for them, at least.
>
> Hi Gene!
>
> Well, it *should* possibly, but most string players have their ears
> set to 12-tET... even though some claim otherwise... or at least
play
> Pythagorean. I believe Paul has some studies to back this up.
>
> Maybe the reason that Neil Haverstick (a very talented musician, by
> the way, if you haven't met him yet) is so successful with 19-tET
is
> the fact that his guitar is FRETTED that way... So he just has to
> learn the notes.
>
> At least that would be *my* guess. That's a far different case
than
> a violinist...
>
> best,
>
> ________ _______ _______
> Joseph Pehrson

Bob:
A lot of early music string players these days are playing pure
thirds. I've always done that from the beginning even in "common
practice" music unless accompanied by 12-tET instruments. Just as the
more "intonationally talented" singers, if I may coin a term, tend
toward just intervals when singing thirds in duet for example, some
string players tend to do the same thing. This has alway been true,
even before the current early music trend came around.

By the way, I used to always tuned my violin in perfect just fifths
(Pythagorean) becuase that's what I was taught to do. This requires
fudging in order to play just intervals, but please don't think good
ears never do this just because you hear common practice recordings
in which they don't. I've seen a lot of generalizations about that in
this list that I strongly disagree with. I can certainly tell the
difference between just and 12-tET thirds and the're not nearly as
rare in string quartet playing, for example, as some here would have
it.

Thank you, Gene! This is valuable input for me. I'm partly sold on 72-
tET because of properties like consistency, etc. It seems to have
them all. On the other hand, it's not terribly clear to me exactly
just how advantageous all these various properties actually are.

I'm very much attracted by 50-tET. Woolhouse really came up with what
I would consider a pretty well-optimized meantone and close ET
equivalent. I like 72-tET precisely because it models the actual
properties of JI more accurately, not glossing them over by
eliminating the syntonic comma, for example. I believe it should be
possible to exploit these properties artistically instead of
eternally viewing them as problems that won't quite go away.

On the 72-tET errors in excess of 7.2, don't you have to go to
compound factors or higher powers of the basic intervals, such as in
the chromatic half steps (24:25 in JI) before you get errors larger
than 7.2 cents? You don't see intervals like that in vertical
structures very often anyway, so I'm not so concerned with a few
cents this way or that in this context. I'm focusing on the harmonic
intervals. I analyzed the JI intervals all the way through taking 9-
19 over 8 and nothing showed up outside of 7.2 cents.

If my method is too crude and is consequently missing important
harmonic intervals, for example I didn't analyze 7:5, then I want to
know that. Thanks again, Gene.

[I wrote:]
>>I'm not quite sure what your subject line means, but I'm glad you
>>enjoy my treatments.

[Joseph Pehrson wrote:]
>Just joking around with you, John... Bravo for your achievements!

Thanks much, Joe!

>You are possibly the most successful "adaptive tuner" in history!
>Seriously.

Well, it won't be long before my work will seem terribly primitive, I'm
sure, but I'm apparently lucky enough to be at the forefront for this
brief moment in time.

>However, you have to publish this stuff in some kind of journal. It
>wouldn't be, necessarily, a "new music" journal since, I feel, this
>subject is of a more "general" nature.

>I would write it up and send it out if I were you, so that you'll
>be "discovered" one of these days!

Please e-mail me off-list with any specific recommendations you might
have. I feel a bit awkward approaching journals that cater to music
PhD's, since my formal education is seriously lacking in that regard.
But... faint heart never won fair maid, so I'm willing to give it a
stab.

JdL

[Bob Wendell wrote:]
>I like 72-tET precisely because it models the actual
>properties of JI more accurately, not glossing them over by
>eliminating the syntonic comma, for example. I believe it should be
>possible to exploit these properties artistically instead of
>eternally viewing them as problems that won't quite go away.

72-tET does, of course, swallow ("gloss over") the Pythagorean comma,
unlike 53-tET. Both leave the syntonic comma problem in place, and
the world still seems to be waiting for musical examples which exploit
this "feature" artistically. To my eye/ear, the challenge seems very
daunting.

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

>Both leave the syntonic comma problem in place, and
> the world still seems to be waiting for musical examples which
exploit
> this "feature" artistically. To my eye/ear, the challenge seems
very
> daunting.

Years ago I wrote some comma ooze pieces, though my favorite was in
the 17-limit and did not involve 81/80. Whether they would be aristic
in your judgment I don't know.

[Gene wrote:]
>Years ago I wrote some comma ooze pieces, though my favorite was in
>the 17-limit and did not involve 81/80. Whether they would be aristic
>in your judgment I don't know.

I don't either! Can we hear them?

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Bob Wendell wrote:]
> >I like 72-tET precisely because it models the actual
> >properties of JI more accurately, not glossing them over by
> >eliminating the syntonic comma, for example. I believe it should
be
> >possible to exploit these properties artistically instead of
> >eternally viewing them as problems that won't quite go away.
>
> 72-tET does, of course, swallow ("gloss over") the Pythagorean
comma,
> unlike 53-tET. Both leave the syntonic comma problem in place, and
> the world still seems to be waiting for musical examples which
exploit
> this "feature" artistically. To my eye/ear, the challenge seems
very
> daunting.
>
> JdL

Hi, John. Yes, we've been over that recently and I agree it does look
daunting from here, but I think someone should try it. Maybe the
default will have to be me, but I hope others more competent will be
inspired to give it a go.

As to 72 glossing over the syntonic comma, I assume you refer to its
reduction to 16.66.... cents and not to its elimination. I would
never use 72 simply because it contains 12-tET as a subset. What a
waste! There are advantages to its containing the 12-tET subset that
I wouldn't want to ignore by any means, but if you go for the quasi-
JI Major third (very close at just under three cents flat), the
second whole step is not 12 steps (ET's version of the whole step),
but 11! Just like JI (almost). Both whole steps (8:9 and 9:10) are
very closely approximated in 72.

If you have or can produce a midi version of it, Gene, I'd love to
hear it. I've never heard anything in which the composer
intentionally exploited the comma.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
>
> >Both leave the syntonic comma problem in place, and
> > the world still seems to be waiting for musical examples which
> exploit
> > this "feature" artistically. To my eye/ear, the challenge seems
> very
> > daunting.
>
> Years ago I wrote some comma ooze pieces, though my favorite was in
> the 17-limit and did not involve 81/80. Whether they would be
aristic
> in your judgment I don't know.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Gene wrote:]

> >Years ago I wrote some comma ooze pieces, though my favorite was
in
> >the 17-limit and did not involve 81/80. Whether they would be
aristic
> >in your judgment I don't know.

> I don't either! Can we hear them?

No, but I could write new ones.

[Bob Wendell wrote:]
>As to 72 glossing over the syntonic comma, I assume you refer to its
>reduction to 16.66.... cents and not to its elimination

Not the syntonic comma, the Pythagorean comma. 72-tET shrinks its
fifths enough that 12 of them come back to the starting point (they're
identical to 12-tET in this regard). Whereas, 53-tET has wider fifths,
extremely accurate, and 12 of them _don't_ come back to the same place.
Both systems hang out the syntonic comma for drift or shift or non-JI
intervallic choices.

JdL

[Gene wrote:]
>>>Years ago I wrote some comma ooze pieces, though my favorite was in
>>>the 17-limit and did not involve 81/80. Whether they would be aristic
>>>in your judgment I don't know.

[I wrote:]
>>I don't either! Can we hear them?

>No, but I could write new ones.

Please do!

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Bob Wendell wrote:]

> >As to 72 glossing over the syntonic comma, I assume you refer to
its
> >reduction to 16.66.... cents and not to its elimination

> Not the syntonic comma, the Pythagorean comma. 72-tET shrinks its
> fifths enough that 12 of them come back to the starting point
(they're
> identical to 12-tET in this regard).

The commas 3^12/2^19 (Pythagorean), 2^6 3^5/5^6 (kleisma), 225/224
(septimal kleisma) and 441/440 (which tells us that 21/20 and 22/21
are being identified) completely characterize the 72-et up to the
11-limit. Other commas of significance are 2401/2400, 4375/4374,
1029/1024, 540/539, 3025/3024, 9801/9800 and 1375/1372.

genewardsmith@juno.com wrote:

> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > Berlin Philharmonic--you know, *the* orchestra.
>
> > They can play in meantone? Or they can't? I forget the context.
>
> They are Big Band traditionalists--tuned sharp, very well-trained,
> but not at all into authenticity.
>
> > 24-tET is pretty much _standard_ in the new music world . . . I'll
> > let Joseph elaborate . . .
>
> When I started looking at this stuff, the first thing I noticed was
> 22, and the second 31, and they still look good to me. I suppose that
> means the "new music world" isn't very mathemically oriented.

With these tunings you will plough a lone furrow. Forgetting about the rest of the "new (ie new
wine in old bottles) music world" and leading by example would seem to be the best way to carry
people along, certainly in my neck of the woods. Very time-consuming and requires vast reserves of
integrity and self-esteem. The expression 'Gardening in a Gale' comes to mind. I speak as a 22-er
and as an aspiriing Blackjacker.

Kind Regards

--- In tuning@y..., genewardsmith@j... wrote:

> The commas 3^12/2^19 (Pythagorean), 2^6 3^5/5^6 (kleisma), 225/224
> (septimal kleisma) and 441/440 (which tells us that 21/20 and 22/21
> are being identified) completely characterize the 72-et up to the
> 11-limit.

While it has been much discussed on the math list, I don't know if
people here are all aware of how shared commas define family
relationships between ets and temperaments. For instance, there is a
kleismic family containing 19, 53 and 72 which share the property
that both the kleisma and the septimal kleisma are unisons. All of
these can be used for the chain-of-minor-thirds system of scales. The
most familiar family of this type is meantone, but the whole picture
is far more involved and multidimensional.

In-Reply-To: <5.1.0.14.1.20011024051440.01fbf040@mail.atl.bellsouth.net>
Oh yes, about this:

[JdL]
> On the planning board: interpreting input pitch bends as defining steps.
> For example, a piece in 19-tET, realized with pitch bends, would be
> understood to represent 19-tET, and adaptive interval adjustments would
> be added overtop of grounding, giving better fifths and major thirds
> than 19-tET does on its own, while preserving the basic identity of the
> 19 steps.

The big win for 19 over 12 is that you can choose between 5- and 7-limit
intervals. But you still have to resolve the comma shifts, which will be
a challenge for a real-time retuner.

[JdL]
> I don't believe that my techniques work "best" on music from the past.
> Several list members use leisure adaptive tuning on their new works.
> However, I'm certainly not going to claim that _all_ music needs to be
> run thru my program! I wouldn't have a clue what to do with a 13-tET
> piece, for example, other than leave it alone. Ditto, I'm guessing, one
> of your Blackjack pieces, Joe.

13-tET and Blackjack are entirely different kettles of ball games. Each
11-limit interval is uniquely represented in Miracle temperament, so you
could set a program to optimize them all as just. Because the shifts are
small, it should work acceptably in real time. It may not give audibly
different results to fixed tuning, but you'll never know until you try.

Graham

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_29265.html#29473
>
> > --- In tuning@y..., genewardsmith@j... wrote:
> > >
> > > I'm waiting for some brave soul to get an orchesta going with
> > strings tuned to sharp fifths and playing in 22-et, myself.
> >
> > I should make this happen within my lifetime . . . starting to
hang
> > out at NEC more . . . maybe I'll even apply and get a
> doctorate . . . of course, a doctorate in a more technical field
> might be more likely to pay for itself . . .
>
>
> Ummm... it's a nice *honor,* Paul, but before you get started in
that
> direction, I would urge you to use your mathmatical abilities to
find
> a high paying "day job" and organize and pay for your *own*
orchestra!
>
> Actually, a gentleman by the name of Eric Grunin has done just that
> in New York. Well, he lost his job as a computer programmer and
had
> angry players running after him for a while... but I hear he may
get
> started with it again. He was, actually, quite successful... big
> reviews in the Times, etc., etc..

Thanks for the advice, Joseph. Ives was a very successful insurance
man, wasn't he?

--- In tuning@y..., BobWendell@t... wrote:

> In the first, fixed frequency case I was saying that the
theoretical
> uncertainty is not really uncertainty at all.

Right -- it's bandwidth.

> Then it doesn't matter
> how large it is compared to the uncertainty intrinsic to the
> mechanics of the perceptual apparatus, does it?

The two are not unrelated. In particular, I was following along with
you when you suggested that, in the amplitude-modulated case, there
could be a perceptual mechanism for focusing in on the _center_ of
the bandwidth, and thereby obtaining a more precise estimate of pitch.
>
> But you're right. As applied to the second stepped frequency case,
> the statement above was not clear. I intended to communicate that
> similarly, whatever APPARENT uncertainties might be introduced by
> bandwidth considerations would not be confusing to the ear in
> determining pitch any more than the other.

How can you be sure? What if locating the center of the bandwidth is
indeed related to how we perceive a well-defined pitch in short-
duration tones?

> FM after all, although
> more complex, still has symmetrical bandwidth products around the
> modulated frequency (or frequencies), regardless of the symmetry or
> asymmetry of the modulating signal.

How is that possible? If the modulating signal is assymmetrical, such
that pitch A, say, the minimum, while another pitch (B) is the
maximum, how can the bandwidth products be symmetrical around _both_
A and B?

Hi John (deLaubenfels),

Over on MakeMicroMusic, Bill Sethares has produced a beautiful piece
of essentially diatonic music where there is

zero retuning motion
zero drift
zero variation in the size of diatonic major seconds
zero variation in the size of diatonic minor seconds

...and...

ZERO HARMONIC PAIN (to my ear at least)!!!

How did he do this? He wrote the piece in 19-tET, and then rendered
it using an acoustic guitar sound, resynthesized to have its partials
exactly in 19-tET.

To my ear, this approach (at least for the acoustic guitar timbre)
seems to work at least as well in producing a sensation of perfect
harmony _and_ perfect melody, as your methods, John. In addition, the
whole question of "limit" becomes moot, because _all_ the partials
are re-mapped to agree with the fixed tuning system.

John, and especially you, Bob Wendell, I would be interested to hear
your reactions to this approach. You can hear it over in the "Files"
section of MakeMicroMusic, under "Sethares". Bob has seemingly been
unwilling to admit the role of eliminating clashing harmonic partials
in producing the sensation of complete concordance . . . but in this
case, they would seem to play a _decisive_ role!

Bill, if you are reading this, I wonder if you can provide us with
some alternate renditions to A/B this with . . . such as 19-tET with
natural timbres, and maybe even 12-tET with natural timbres and 12-
tET with 12-tET timbres.

I know that if the partials are remapped _too far_ from a harmonic
series, the integrity of the pitches themselves is lost . . . but in
this case, they're only moved by a few cents . . . I wonder how far
this can be taken?

--- In tuning@y..., BobWendell@t... wrote:

> On the 72-tET errors in excess of 7.2, don't you have to go to
> compound factors or higher powers of the basic intervals, such as
in
> the chromatic half steps (24:25 in JI) before you get errors larger
> than 7.2 cents?

If your set of basic intervals is the 17-limit, then you're correct.
But in the 19-limit, you'll see that, if you try to play a chord like
8:13:19, you'll have to commit an error greater than half a step of
72-tET (i.e., 8.333 cents) in at least one of the intervals.

> You don't see intervals like that in vertical
> structures very often anyway, so I'm not so concerned with a few
> cents this way or that in this context.

You shouldn't be.

> I'm focusing on the harmonic
> intervals. I analyzed the JI intervals all the way through taking 9-
> 19 over 8 and nothing showed up outside of 7.2 cents.
>
> If my method is too crude and is consequently missing important
> harmonic intervals, for example I didn't analyze 7:5, then I want
to
> know that.

You evidently didn't analyze 19:13, and if you did, you didn't
enforce consistency.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> 72-tET does, of course, swallow ("gloss over") the Pythagorean
comma,
> unlike 53-tET. Both leave the syntonic comma problem in place, and
> the world still seems to be waiting for musical examples which
exploit
> this "feature" artistically.

I recall a piece on Carl Lumma's copy of the JIN tape which did so.
Unfortunately, JIN never sent me the tape, even though I sent them a
check many years ago.

--- In tuning@y..., BobWendell@t... wrote:
> --- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> > 72-tET does, of course, swallow ("gloss over") the Pythagorean
> comma,
> > unlike 53-tET. Both leave the syntonic comma problem in place,
[...]
> >
> > JdL

>
> As to 72 glossing over the syntonic comma, I assume you refer to
its
> reduction to 16.66.... cents and not to its elimination.

Read the above again: John deLaubenfels said that 72 glosses over or
swallows the _Pythagorean_ comma -- not the _syntonic_ comma.

I would
> never use 72 simply because it contains 12-tET as a subset. What a
> waste!

I don't understand this statement. Wouldn't being able to build upon
12-tET and 24-tET notation, and the wide world of musicians who have
been trained in them, be an advantage?

--- In tuning@y..., BobWendell@t... wrote:

> If you have or can produce a midi version of it, Gene, I'd love to
> hear it. I've never heard anything in which the composer
> intentionally exploited the comma.

Neil Haverstick said he has done so in 34-tET. I wonder if he can
point to any examples?

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <5.1.0.14.1.20011024051440.01fbf040@m...>
> Oh yes, about this:
>
> [JdL]
> > On the planning board: interpreting input pitch bends as defining
steps.
> > For example, a piece in 19-tET, realized with pitch bends, would
be
> > understood to represent 19-tET, and adaptive interval adjustments
would
> > be added overtop of grounding, giving better fifths and major
thirds
> > than 19-tET does on its own, while preserving the basic identity
of the
> > 19 steps.
>
> The big win for 19 over 12 is that you can choose between 5- and 7-
limit
> intervals. But you still have to resolve the comma shifts, which
will be
> a challenge for a real-time retuner.

What comma shifts are you referring to? 19-tET of course
swallows/glosses over the usual syntonic comma.

[Paul E wrote:]
>Hi John (deLaubenfels),

>Over on MakeMicroMusic, Bill Sethares has produced a beautiful piece
>of essentially diatonic music where there is

>zero retuning motion
>zero drift
>zero variation in the size of diatonic major seconds
>zero variation in the size of diatonic minor seconds

>...and...

>ZERO HARMONIC PAIN (to my ear at least)!!!

>How did he do this? He wrote the piece in 19-tET, and then rendered
>it using an acoustic guitar sound, resynthesized to have its partials
>exactly in 19-tET.

Very clever, that Bill! The timbre sounds perfectly fine to me, and
like you, I find the chords to be perfectly in tune. I don't happen
to like the piece (the heavy attack and uneven volume of the notes makes
me nervous, a sensation I get from a lot of solo guitar music).

>To my ear, this approach (at least for the acoustic guitar timbre)
>seems to work at least as well in producing a sensation of perfect
>harmony _and_ perfect melody, as your methods, John. In addition, the
>whole question of "limit" becomes moot, because _all_ the partials
>are re-mapped to agree with the fixed tuning system.

Yes... certainly one can't achieve, say, the sound of a dom 9 chord in
this timbre, which (to take one example) I might miss, but I am very
impressed by this technique and eager to experience more of its
possibilities. I don't see any reason why someone like Bill couldn't do
up an entire sound font of all 128 General MIDI instruments with
remapped partials for 19-tET (and other sets for other ET's). A whole
new world of well-tuned music might open up!

>John, and especially you, Bob Wendell, I would be interested to hear
>your reactions to this approach. You can hear it over in the "Files"
>section of MakeMicroMusic, under "Sethares". Bob has seemingly been
>unwilling to admit the role of eliminating clashing harmonic partials
>in producing the sensation of complete concordance . . . but in this
>case, they would seem to play a _decisive_ role!

Agreed. To the extent that a fixed tuning works, this is definitely
superior to my adaptive methods, in which there are always some
horizontal prices (except in the imperfect COFT calculations, of
course).

>Bill, if you are reading this, I wonder if you can provide us with
>some alternate renditions to A/B this with . . . such as 19-tET with
>natural timbres, and maybe even 12-tET with natural timbres and 12-
>tET with 12-tET timbres.

I'd love to get ahold of a MIDI file of this, to run thru my own
methods using harmonic guitar voice(s), for comparison.

>I know that if the partials are remapped _too far_ from a harmonic
>series, the integrity of the pitches themselves is lost . . . but in
>this case, they're only moved by a few cents . . . I wonder how far
>this can be taken?

Good question. In Bill's vivid .wav files from the CD companion to
his book "Tuning, Timbre, Spectrum, Scale", 2.1 stretched octave
overtones don't sound real bad. They might represent an approximate
outer limit for overtone stretch (or compression), with fifths
proportionately narrower in outer limit range, perhaps. The corrections
for 19-tET are much less than this, IIRC.

To get a better idea of the effectiveness of this kind of technique, I'd
like to have an hour or more of music I really love, to try to sense
whether the ear over time becomes fatigued with these slightly fudged
overtones (fudged compared to harmonic, that is...).

Bill seems to have his finger on some really exciting possibilities.
Hope he continues to develop them!!

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> Yes... certainly one can't achieve, say, the sound of a dom 9 chord
in
> this timbre, which (to take one example) I might miss,

Hmm . . . what exactly do you mean? And what about a dom 9 chord in
12-tET, with all the partials mapped to 12-tET? How do you know you
wouldn't like that as much as you like a just 7-limit dom 9 chord
with natural timbres?

> but I am very
> impressed by this technique and eager to experience more of its
> possibilities. I don't see any reason why someone like Bill
couldn't do
> up an entire sound font of all 128 General MIDI instruments with
> remapped partials for 19-tET (and other sets for other ET's). A
whole
> new world of well-tuned music might open up!

This was one of the good ideas that the "nutty professor" was
actually advocating some years ago, before he completely fell off the
deep end . . .
>
> Good question. In Bill's vivid .wav files from the CD companion to
> his book "Tuning, Timbre, Spectrum, Scale", 2.1 stretched octave
> overtones don't sound real bad. They might represent an approximate
> outer limit for overtone stretch (or compression),

That's about a 5% stretch.

> with fifths
> proportionately narrower in outer limit range, perhaps.

Unclear.

> The corrections
> for 19-tET are much less than this, IIRC.

In the 5-limit (as appropriate to Bill's guitar piece), 19-tET has a
maximum deviation from JI of 0.43%.
>
> To get a better idea of the effectiveness of this kind of
technique, I'd
> like to have an hour or more of music I really love, to try to sense
> whether the ear over time becomes fatigued with these slightly
fudged
> overtones (fudged compared to harmonic, that is...).

Me too.

> Bill seems to have his finger on some really exciting possibilities.
> Hope he continues to develop them!!

Amen!

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > In the first, fixed frequency case I was saying that the
> theoretical
> > uncertainty is not really uncertainty at all.
>
> Right -- it's bandwidth.
>
> > Then it doesn't matter
> > how large it is compared to the uncertainty intrinsic to the
> > mechanics of the perceptual apparatus, does it?
>
> The two are not unrelated. In particular, I was following along
with
> you when you suggested that, in the amplitude-modulated case, there
> could be a perceptual mechanism for focusing in on the _center_ of
> the bandwidth, and thereby obtaining a more precise estimate of
pitch.

Bob now:
Yes, you have it!

> >Bob before again:
> > But you're right. As applied to the second stepped frequency
case,
> > the statement above was not clear. I intended to communicate that
> > similarly, whatever APPARENT uncertainties might be introduced by
> > bandwidth considerations would not be confusing to the ear in
> > determining pitch any more than the other.
>

Paul:
> How can you be sure? What if locating the center of the bandwidth
is
> indeed related to how we perceive a well-defined pitch in short-
> duration tones?
>
Bob now:
I believe it is.

> > FM after all, although
> > more complex, still has symmetrical bandwidth products around the
> > modulated frequency (or frequencies), regardless of the symmetry
or
> > asymmetry of the modulating signal.
>
> How is that possible? If the modulating signal is assymmetrical,
such
> that pitch A, say, the minimum, while another pitch (B) is the
> maximum, how can the bandwidth products be symmetrical around
_both_
> A and B?

Bob now:
It seems that maybe you're confusing the frequency of the successive
tones with the modulating signal. They are in fact the *modulated*
signal. If I alternate the pitch of a continuous tone every half
second between two frequencies, the continuous tone is being
*modulated* by a *modulating* signal that is a square wave at a
frequency of two cycles per second. Even if we increase that
*modulating* frequency to 30 Hz it is still a very low frequency even
though much higher than most musicians would ever be called upon to
finger. (Can you trill at 30 Hz?)

In traditional FM radio technology you would never see a signal (our
continuous tone of alternating pitch) modulated with a signal that
varied it by such a huge percentage of itself. If we nonetheless view
our alternating musical tone as a conventional FM signal with a
radically huge modulation from a center carrier, it would have a
center frequency at the arithmetic mean of the two pitches.

I don't see this as a very intuitively valuable way to analyze this
situation, however, since our ears absolutely DO NOT perceive these
low-frequency moulations as deviations from a center frequency but as
discrete, consecutive pithes. Indulging ourselves in spite of all,
conventional FM radio sidebands are generated by a modulating signal
that is low in frequency compared to the amount of frequency
deviation from a center carrier frequency in contrast to our trill
model. The square wave harmonics of the modulating signal would need
to be about 11 times the fundamental (odd harmonics) to produce a
good square wave approximation, so the maximum modulating frequency
might be around 110 Hz or more at an amplitude of 1/11 times the
fundamental for a trill at 10 Hz (much more realistic than 30 Hz).

Again, I'm not familiar with the mathematics involving Bessel
functions that are used to predict the sidebands, but I know the
sidebands depend on something called the modulation index, which is
the ratio of the frequency deviation from center to the bandwidth of
the modulating signal. (Boy, I'm really starting to stretch my memory
here. I haven't been an FM radio engineer since 1981.)

In our extreme situation I believe I recall that the modulation index
would be low, since the frequency deviation from center, although
unusually large as a percentage of the modulated frequency, would be
small for stepwise intervals compared to the highest modulating
frequency (110 Hz for a fairly fast trill).

It would become larger for large intervals, but still low and who
trills through fifths? Arpeggios would involve a stepped waveform
rather than a square wave, but that would not effect the frequency
analysis signficantly and would up the ante a bit on the modulation
index. However, when we move away from our trill model, the analysis
becomes tremendously complicated and way outside the bounds of FM
theory I'm familiar with, since even defining a center frequency and
therefore frequency deviation from it becomes impossible.

When the modulation index is low, the ultimate bandwidth of the
modulated signal itself remains within the bounds of the bandwidth of
the modulating signal. This is the case in our trill model. This
analysis has been on the fundamental of the modulated signal, but
would apply equally to its partials. The partials would multiply the
modulation index by a factor equal to its harmonic order.
Nonetheless, these are low indices for FM.

Bear in mind, however, that this ultimate bandwidth of the modulated
signal amounts to nothing more than a description of the frequency
components that switch the frequency of the tones, and they are still
symmetrical about the center frequency and should not obscure pitch
perception at any frequency low enough to be executable by mortal
human musicians.

At much higher frequencies of the trill, I would expect the ear to
start hearing the wave complex generated as a modulation of a center
frequency and start getting confused, but this would have to be at
quite a high rate and would not be preceived as a mere shift in the
two pitches, but a whole different kind of sound with competely new
and alien pitch components. The mere fact that we're not hearing that
indicates to me that this is not introducing pitch indeterminacy.

Finally, intuitively I can't see how alternating tones by sweeping
the frequency quickly between two straight pitches of relatively long
duration vis-a-vis that of the quick sweep from one to the other and
at such a low frequency could introduce any significant indeterminacy
in pitch perception. This is the best i can do. Sorry if it is not
satisfactory. Maybe someone more knowledgeable in this area can chip
in?

[I wrote:]
>>Yes... certainly one can't achieve, say, the sound of a dom 9 chord in
>>this timbre, which (to take one example) I might miss...

[Paul:]
>Hmm . . . what exactly do you mean? And what about a dom 9 chord in
>12-tET, with all the partials mapped to 12-tET? How do you know you
>wouldn't like that as much as you like a just 7-limit dom 9 chord
>with natural timbres?

I don't know, because I've never heard that. Just idle speculation at
this point. Isn't that what the Hammond (?) organ does/did? Most
people say it sounds a bit cheesy, but I don't know if it's for that
reason.

[JdL:]
>>but I am very impressed by this technique and eager to experience more
>>of its possibilities. I don't see any reason why someone like Bill
>>couldn't do up an entire sound font of all 128 General MIDI
>>instruments with remapped partials for 19-tET (and other sets for
>>other ET's). A whole new world of well-tuned music might open up!

[Paul:]
>This was one of the good ideas that the "nutty professor" was
>actually advocating some years ago, before he completely fell off the
>deep end . . .

No kidding! I've given up on him lately, 'cause he doesn't choose to
stop referring to me as "that liar, John deLaubenfels." Glad to hear
that he's had a good idea or two, though.

[JdL:]
>>Good question. In Bill's vivid .wav files from the CD companion to
>>his book "Tuning, Timbre, Spectrum, Scale", 2.1 stretched octave
>>overtones don't sound real bad. They might represent an approximate
>>outer limit for overtone stretch (or compression),

[Paul:]
>That's about a 5% stretch.

I'm thinking of octaves per octave, logarithmically. Thus, 2.1 is
1284.47 cents, so that represents a stretch ratio of 1284.47/1200.00 =~
1.0704 . So, applied to a fifth, at 701.95 cents harmonic, that'd map
to about +/- 50 cents. Yes? Which is _huge_ compared to the correction
for 19-tET.

[JdL:]
>>with fifths proportionately narrower in outer limit range, perhaps.

[Paul:]
>Unclear.

See above.

[JdL:]
>>To get a better idea of the effectiveness of this kind of technique,
>>I'd like to have an hour or more of music I really love, to try to
>>sense whether the ear over time becomes fatigued with these slightly
>>fudged overtones (fudged compared to harmonic, that is...).

[Paul:]
>Me too.

[JdL:]
>>Bill seems to have his finger on some really exciting possibilities.
>>Hope he continues to develop them!!

[Paul:]
>Amen!

Go, Bill!

JdL

--- In tuning@y..., BobWendell@t... wrote:

> > The two are not unrelated. In particular, I was following along
> with
> > you when you suggested that, in the amplitude-modulated case,
there
> > could be a perceptual mechanism for focusing in on the _center_
of
> > the bandwidth, and thereby obtaining a more precise estimate of
> pitch.
>
> Bob now:
> Yes, you have it!

Bob, I was following along with you on this all along! No different
now.
>
> > >Bob before again:
> > > But you're right. As applied to the second stepped frequency
> case,
> > > the statement above was not clear. I intended to communicate
that
> > > similarly, whatever APPARENT uncertainties might be introduced
by
> > > bandwidth considerations would not be confusing to the ear in
> > > determining pitch any more than the other.
> >
>
> Paul:
> > How can you be sure? What if locating the center of the bandwidth
> is
> > indeed related to how we perceive a well-defined pitch in short-
> > duration tones?
> >
> Bob now:
> I believe it is.

Then we should be able to apply it to the case where the short pitch
is preceded and followed by different frequencies, rather than by
silence.
>
> > > FM after all, although
> > > more complex, still has symmetrical bandwidth products around
the
> > > modulated frequency (or frequencies), regardless of the
symmetry
> or
> > > asymmetry of the modulating signal.
> >
> > How is that possible? If the modulating signal is assymmetrical,
> such
> > that pitch A, say, the minimum, while another pitch (B) is the
> > maximum, how can the bandwidth products be symmetrical around
> _both_
> > A and B?
>
> Bob now:
> It seems that maybe you're confusing the frequency of the
successive
> tones with the modulating signal. They are in fact the *modulated*
> signal.

I understand that, and don't think I was confused about anything.

> If I alternate the pitch of a continuous tone every half
> second between two frequencies, the continuous tone is being
> *modulated* by a *modulating* signal that is a square wave at a
> frequency of two cycles per second. Even if we increase that
> *modulating* frequency to 30 Hz it is still a very low frequency
even
> though much higher than most musicians would ever be called upon to
> finger. (Can you trill at 30 Hz?)

I can trill at 25Hz on the guitar, by tapping the upper note
alternately with my right hand index finger and my left hand ring
finger -- each must tap a little more than 6 times per second. But I
don't think trills are too relevant here.
>
> In traditional FM radio technology you would never see a signal
(our
> continuous tone of alternating pitch) modulated with a signal that
> varied it by such a huge percentage of itself.

Well then forget about traditional FM radio technology! Clearly it's
not relevant here.

> If we nonetheless view
> our alternating musical tone as a conventional FM signal with a
> radically huge modulation from a center carrier, it would have a
> center frequency at the arithmetic mean of the two pitches.

Right -- but what's the _perceptual uncertainty_ of _each_ of these
pitches???
>
> I don't see this as a very intuitively valuable way to analyze this
> situation, however, since our ears absolutely DO NOT perceive these
> low-frequency moulations as deviations from a center frequency but
as
> discrete, consecutive pithes.

Exactly!
>
> It would become larger for large intervals, but still low and who
> trills through fifths?

What if we're talking about a 1 Hz "trill"? Or just a simple I-V-I
progression?
>
> Bear in mind, however, that this ultimate bandwidth of the
modulated
> signal amounts to nothing more than a description of the frequency
> components that switch the frequency of the tones, and they are
still
> symmetrical about the center frequency

In this case, a quarter-tone away from the third!

> and should not obscure pitch
> perception at any frequency low enough to be executable by mortal
> human musicians.

I have to say you completely missed the point. Oh well, I'm
patient . . .
>
> At much higher frequencies of the trill, I would expect the ear to
> start hearing the wave complex generated as a modulation of a
center
> frequency and start getting confused, but this would have to be at
> quite a high rate and would not be preceived as a mere shift in the
> two pitches, but a whole different kind of sound with competely new
> and alien pitch components.

This was demonstrated in Harold Fortuin's first piece that he
presented at NEC on Friday, where notes succeeded one another at
accelerating and decelarating rates, often as fast as 50 per second
or faster.
>
> Finally, intuitively I can't see how alternating tones by sweeping
> the frequency quickly between two straight pitches of relatively
long
> duration vis-a-vis that of the quick sweep from one to the other
and
> at such a low frequency could introduce any significant
indeterminacy
> in pitch perception.

My point is that the bandwidth relative to _each_ of the pitches is
assymmetrical, hence the argument you used to argue the irrelevance
of the classical uncertainty principle above (and that I've been
following you on all along) wouldn't be able to apply to this
situation as well.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [I wrote:]
> >>Yes... certainly one can't achieve, say, the sound of a dom 9
chord in
> >>this timbre, which (to take one example) I might miss...
>
> [Paul:]
> >Hmm . . . what exactly do you mean? And what about a dom 9 chord
in
> >12-tET, with all the partials mapped to 12-tET? How do you know
you
> >wouldn't like that as much as you like a just 7-limit dom 9 chord
> >with natural timbres?
>
> I don't know, because I've never heard that. Just idle speculation
at
> this point. Isn't that what the Hammond (?) organ does/did?

Sort of. The tonebars on the Hammond organ did, at least at one
point, engage "overtones" that were within 1 cent of 12-tET. However,
each of these overtones, and the fundamental itself, were not quite a
pure sine wave -- and the typical distortion added by the speaker
would create further natural harmonics.

> Most
> people say it sounds a bit cheesy,

The Hammond organ is one of my favorite sounds in the world,
especially though a rotating leslie speaker (mine was destroyed in a
fire :( ). What do _you_ think?

> but I don't know if it's for that
> reason.

Define "cheesy". I've been told I'm cheesy.

> No kidding! I've given up on him lately, 'cause he doesn't choose
to
> stop referring to me as "that liar, John deLaubenfels."

He's obviously decided that it's better to stick you in some kind of
bin so that he can dismiss you (as he does to everyone) rather than
actually taking the trouble to understand what you're doing.

> Glad to hear
> that he's had a good idea or two, though.

Not really his own idea, but sure. You can find some of his
more "sane" stuff archived on the Internet. Unfortunately, there was
always a high proportion of deceptively plausible, yet incorrect,
material in his writings.

> [JdL:]
> >>with fifths proportionately narrower in outer limit range,
perhaps.
>
> [Paul:]
> >Unclear.
>
> See above.

Well, I mean it's unclear whether this truly can function as a
practical rule for an "outer limit range" for retuning the 3rd
partial, or 6th partial, or whatever. One would have to do a wide
range of experiments to really decide.

[I wrote:]
>>I don't know, because I've never heard that. Just idle speculation at
>>this point. Isn't that what the Hammond (?) organ does/did?

[Paul:]
>Sort of. The tonebars on the Hammond organ did, at least at one
>point, engage "overtones" that were within 1 cent of 12-tET. However,
>each of these overtones, and the fundamental itself, were not quite a
>pure sine wave -- and the typical distortion added by the speaker
>would create further natural harmonics.

Hmmm: that will continue to be the case with any experiments of this
nature, that harmonic distortion in less-than-perfect components will
cause more trouble than it does with instruments of harmonic timbres.
I run my computer output through my best stereo system (old, but not too
bad) rather than cheap computer speakers, so perhaps that helps with
Bill's piece.

[JdL:]
>>Most people say it sounds a bit cheesy,

[Paul:]
>The Hammond organ is one of my favorite sounds in the world,
>especially though a rotating leslie speaker (mine was destroyed in a
>fire :( ). What do _you_ think?

Not sure I've ever heard one! Any sources for recordings?

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [I wrote:]
> >>I don't know, because I've never heard that. Just idle
speculation at
> >>this point. Isn't that what the Hammond (?) organ does/did?
>
> [Paul:]
> >Sort of. The tonebars on the Hammond organ did, at least at one
> >point, engage "overtones" that were within 1 cent of 12-tET.
However,
> >each of these overtones, and the fundamental itself, were not
quite a
> >pure sine wave -- and the typical distortion added by the speaker
> >would create further natural harmonics.
>
> Hmmm: that will continue to be the case with any experiments of this
> nature, that harmonic distortion in less-than-perfect components
will
> cause more trouble than it does with instruments of harmonic
timbres.

Well, this distortion tends to be _intentional_ when it comes to the
Hammond/Leslie combination.

> I run my computer output through my best stereo system (old, but
not too
> bad) rather than cheap computer speakers, so perhaps that helps with
> Bill's piece.

It can't hurt, especially for experiments of this nature!
>
> Not sure I've ever heard one!

Are you quite serious?

> Any sources for recordings?

Anything with Jimmy Smith on it (including two fine albums with Wes
Montgomery), or Joey deFrancesco (a great Jimmy Bruno album and a
great John McLaughlin album come to mind), or the many other Hammond
virtuosos out there (as you can see I'm more focused on guitarists).

--- In tuning@y..., BobWendell@t... wrote:

/tuning/topicId_29265.html#29515

> If you have or can produce a midi version of it, Gene, I'd love to
> hear it. I've never heard anything in which the composer
> intentionally exploited the comma.
>

I believe David Beardsley has a piece like this. At least, David
*thought* he was exploiting the comma, but Paul Erlich seemed to feel
the interval he used was a lot larger than that...

Oh well... it was quite a nice piece anyway.

Maybe it's on here:

http://artists.mp3s.com/artists/200/david_beardsley1.html

What's with the brick background?? I feel like I'm in the middle of
West Side Story... :) [actually that seems pretty peaceful right
now...]

_________ ________ ________
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> /tuning/topicId_29265.html#29515
>
>
> > If you have or can produce a midi version of it, Gene, I'd love
to
> > hear it. I've never heard anything in which the composer
> > intentionally exploited the comma.
> >
>
> I believe David Beardsley has a piece like this. At least, David
> *thought* he was exploiting the comma, but Paul Erlich seemed to
feel
> the interval he used was a lot larger than that...

Either way, it wasn't the syntonic comma, which is what we mean
by "the comma".

--- In tuning@y..., genewardsmith@j... wrote:

/tuning/topicId_29265.html#29528

> --- In tuning@y..., genewardsmith@j... wrote:
>
> > The commas 3^12/2^19 (Pythagorean), 2^6 3^5/5^6 (kleisma),
225/224
> > (septimal kleisma) and 441/440 (which tells us that 21/20 and
22/21
> > are being identified) completely characterize the 72-et up to the
> > 11-limit.
>
> While it has been much discussed on the math list, I don't know if
> people here are all aware of how shared commas define family
> relationships between ets and temperaments. For instance, there is
a kleismic family containing 19, 53 and 72 which share the property
> that both the kleisma and the septimal kleisma are unisons. All of
> these can be used for the chain-of-minor-thirds system of scales.
The most familiar family of this type is meantone, but the whole
picture is far more involved and multidimensional.

Hello Gene!

If I'm not mistaken, this is what Paul Erlich likes to talk about
quite a bit of the time.... how eliminating *one* comma can create a
meantone and eliminating *another* comma in another "dimension"
creates an ET...

This seems to be pretty nicely illustrated in Paul's terrific _Forms
of Tonality_ if I'm understanding it correctly...

Thanks, by the way, for bringing some of the "lighter" math
discussions over here. I always *intend* to study the tuning-math
list but never seem to quite get to it... (is that "self-protection"
or what! :)

_________ _______ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> > While it has been much discussed on the math list, I don't know
if
> > people here are all aware of how shared commas define family
> > relationships between ets and temperaments. For instance, there
is
> a kleismic family containing 19, 53 and 72 which share the property
> > that both the kleisma and the septimal kleisma are unisons. All
of
> > these can be used for the chain-of-minor-thirds system of scales.
> The most familiar family of this type is meantone, but the whole
> picture is far more involved and multidimensional.
>
> Hello Gene!
>
> If I'm not mistaken, this is what Paul Erlich likes to talk about
> quite a bit of the time.... how eliminating *one* comma can create
a
> meantone and eliminating *another* comma in another "dimension"
> creates an ET...

Actually, Gene is talking about a different, but related, notion.
Rather than a multidimensional space where each point represents a
different pitch, Gene is referring to a multidimensional space where
each point represents a different precise tuning system. To begin to
understand what is involved here, one might want to look at Herman
Miller's plot of ETs on a 2-dimensional plane, with one axis
representing the error in the major third, and the other axis
representing the error in the fifth. What you see is that the points
form straight lines, criss-crossing the graph in many directions. One
of the lines connects all the meantone ETs (26, 19, 50, 31, 43, 55,
12); one of the lines connects all the schismic ETs; one of the lines
connects all the diaschismic ETs, etc.

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

/tuning/topicId_29265.html#29548

> Thanks for the advice, Joseph. Ives was a very successful insurance
> man, wasn't he?

And Borodin was a chemist... so there are lots of examples. You
could be the first "astro-physicist" to conduct an orchestra,
however... Go for it!

________ _______ ________
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_29265.html#29548
>
> > Thanks for the advice, Joseph. Ives was a very successful
insurance
> > man, wasn't he?
>
> And Borodin was a chemist... so there are lots of examples. You
> could be the first "astro-physicist" to conduct an orchestra,
> however... Go for it!

I'm not very "astro" (unless you mean "spacy"), and I know next to
nothing about conducting . . . though you might say I "conduct" some
of my band's improvisations a bit using body motions and facial
expressions . . . anyway, further discussion on this, if any, belongs
_off-list_ . . .

In-Reply-To: <9r9tho+rgl4@eGroups.com>
Paul asked

> What comma shifts are you referring to? 19-tET of course
> swallows/glosses over the usual syntonic comma.

I meant that, because 19-tET glosses over them, the adaptive tuning
algorithm has a lot more work to do. If you're playing in Blackjack, the
whole problem magically disappears.

BTW, I suggested 19-equal was 7-limit unique, which is wrong. Still, an
algorithm using a meantone first approximation should be able to tell its
8:7 from its 7:6.

Graham

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > > The two are not unrelated. In particular, I was following along
> > with
> > > you when you suggested that, in the amplitude-modulated case,
> there
> > > could be a perceptual mechanism for focusing in on the _center_
> of
> > > the bandwidth, and thereby obtaining a more precise estimate of
> > pitch.
> >
> > Bob:
> > Yes, you have it!
>
> Bob, I was following along with you on this all along! No different
> now.
> >
Bob now: Didn't mean to imply otherwise. You have it, that's all.
Nothing about when you got it.

> > > >Bob before again:
> > > > But you're right. As applied to the second stepped frequency
> > case,
> > > > the statement above was not clear. I intended to communicate
> that
> > > > similarly, whatever APPARENT uncertainties might be
introduced
> by
> > > > bandwidth considerations would not be confusing to the ear in
> > > > determining pitch any more than the other.
> > >
> >
> > Paul:
> > > How can you be sure? What if locating the center of the
bandwidth
> > is
> > > indeed related to how we perceive a well-defined pitch in short-
> > > duration tones?
> > >
> > Bob:
> > I believe it is.
>
> Then we should be able to apply it to the case where the short
pitch
> is preceded and followed by different frequencies, rather than by
> silence.
> >
> > > > FM after all, although
> > > > more complex, still has symmetrical bandwidth products around
> the
> > > > modulated frequency (or frequencies), regardless of the
> symmetry
> > or
> > > > asymmetry of the modulating signal.
> > >
> > > How is that possible? If the modulating signal is
assymmetrical,
> > such
> > > that pitch A, say, the minimum, while another pitch (B) is the
> > > maximum, how can the bandwidth products be symmetrical around
> > _both_
> > > A and B?
> >
> > Bob:
> > It seems that maybe you're confusing the frequency of the
> successive
> > tones with the modulating signal. They are in fact the
*modulated*
> > signal.
>
> I understand that, and don't think I was confused about anything.
>
Bob now:
Well, neither would be a center frequency in the case I analyzed
(square wave modulation of a center frequency at the arithmetic mean.)

Bob before:
> > If I alternate the pitch of a continuous tone every half
> > second between two frequencies, the continuous tone is being
> > *modulated* by a *modulating* signal that is a square wave at a
> > frequency of two cycles per second. Even if we increase that
> > *modulating* frequency to 30 Hz it is still a very low frequency
> even
> > though much higher than most musicians would ever be called upon
to
> > finger. (Can you trill at 30 Hz?)
>

> I can trill at 25Hz on the guitar, by tapping the upper note
> alternately with my right hand index finger and my left hand ring
> finger -- each must tap a little more than 6 times per second. But
I
> don't think trills are too relevant here.

Bob now:
Well, that's 12 Hz since each note is only half the cycle.

The trill is the only case I can reduce to a square wave modulation
of a center frequency at the arithmetic mean. I'm trying to posit the
simplest case that would demonstrate principle accurately, since I'm
not mathematically competent to analyze the more complex case of
stepped modulation that would have no definable center frequency.
> >
Bob before:
> > In traditional FM radio technology you would never see a signal
> (our
> > continuous tone of alternating pitch) modulated with a signal
that
> > varied it by such a huge percentage of itself.
>
> Well then forget about traditional FM radio technology! Clearly
it's
> not relevant here.
>

Bob now:
It seems to be relevant in the simple case of the trill. Again,
that's why I reduced the problem to this simple case. The fundamental
principle affecting determinacy should not change.

> > If we nonetheless view
> > our alternating musical tone as a conventional FM signal with a
> > radically huge modulation from a center carrier, it would have a
> > center frequency at the arithmetic mean of the two pitches.
>
> Right -- but what's the _perceptual uncertainty_ of _each_ of these
> pitches???
> >
> > I don't see this as a very intuitively valuable way to analyze
this
> > situation, however, since our ears absolutely DO NOT perceive
these
> > low-frequency moulations as deviations from a center frequency
but
> as
> > discrete, consecutive pithes.
>
> Exactly!
> >
> > It would become larger for large intervals, but still low and who
> > trills through fifths?
>
> What if we're talking about a 1 Hz "trill"? Or just a simple I-V-I
> progression?
> >
Bob now:
The modulating freuqncy at 1 Hz is so low, it more than compensates
for the increase in frequency deviation.

Bob before:
> > Bear in mind, however, that this ultimate bandwidth of the
> modulated
> > signal amounts to nothing more than a description of the
frequency
> > components that switch the frequency of the tones, and they are
> still
> > symmetrical about the center frequency
>
> In this case, a quarter-tone away from the third!
>
> > and should not obscure pitch
> > perception at any frequency low enough to be executable by mortal
> > human musicians.
>
> I have to say you completely missed the point. Oh well, I'm
> patient . . .
>
Why? I think this simple case is perfectly extensible to the stepped
modulation representing a scale or arpeggio. What difference could it
make in principle that we went back to the starting pitch instead of
to a third pitch? The point is that nothing audibly different is
introduced by this modulation scheme until we reach a modulating
frequency high enough to start generating musically spurious
frequencies that have nothing to do with the two original tones
between which we were switching, so if we don't hear that, we're
doing ok on our determinacy.
>

> > At much higher frequencies of the trill, I would expect the ear
to
> > start hearing the wave complex generated as a modulation of a
> center
> > frequency and start getting confused, but this would have to be
at
> > quite a high rate and would not be preceived as a mere shift in
the
> > two pitches, but a whole different kind of sound with competely
new
> > and alien pitch components.
>
> This was demonstrated in Harold Fortuin's first piece that he
> presented at NEC on Friday, where notes succeeded one another at
> accelerating and decelarating rates, often as fast as 50 per second
> or faster.
> >
Precisely, but this is never a practical problem in normal
performance, is it? If you deliberately introduce it in an electric
composition, then you wanted it, right? so where' the problem?

> > Finally, intuitively I can't see how alternating tones by
sweeping
> > the frequency quickly between two straight pitches of relatively
> long
> > duration vis-a-vis that of the quick sweep from one to the other
> and
> > at such a low frequency could introduce any significant
> indeterminacy
> > in pitch perception.
>
Paul:
> My point is that the bandwidth relative to _each_ of the pitches is
> assymmetrical, hence the argument you used to argue the irrelevance
> of the classical uncertainty principle above (and that I've been
> following you on all along) wouldn't be able to apply to this
> situation as well.

Bob now:
Yes, and my point is that this *a*symmetry with respect to each tone
is in contrast *symmetrical* about their arithmetic mean and hence
not directly related musically to the tones themselves. These
sidebands are only perceptiple in terms of their switching effect
until the modulating frequency increases to such a point that
musically spurious tones become manifest to the human ear. You may
view this as a crude argument for backing up my intuitive response to
your question, but it's the best I can do.

Empirally speaking, we can perceive extremely high rates of
arpeggiation of just chords and recognize their purity or lack of it
without a problem. If this "classical indeterminacy" issue were a
real problem, how could that be possible?

Paul:
> Anything with Jimmy Smith on it (including two fine albums with Wes
> Montgomery), or Joey deFrancesco (a great Jimmy Bruno album and a
> great John McLaughlin album come to mind), or the many other
Hammond
> virtuosos out there (as you can see I'm more focused on guitarists).

Bob:
Yeah, the old Hammond/Leslies are a great sound for jazz and
especially bluesy funk of the Jimmy Smith genre. I've been a fan of
both Smith and Montgomery for decades. (Sacred music of the European
masters is the most uplifting music for me, but I like to "git down"
as much as anybody and good, powerful black gospel knocks me out,
too!)

Paul, I heard the "Truth on a Bus" file at Sethares' site long ago.
I'd like to hear more plain, unSetharized 19-tET to see how much I
like it compared to 12-tET with normal harmonic timbres, though.

I find Sethares' work fascinating. I called up my gifted violinist
sister and got her to listen to "Turqoise Dabo Girl" in 10-tET. She
flipped out. Said the "Star Wars bar scene" should have used that
music instead of what they did use. I truthfully would not have
predicted that such tunings could sound as consonant as they do by
simply playing iwth harmonic content.

However, in practice I have to deal with the human voice, and there
we have no slack on such partial manipulation. I happen to think the
human voice is the most beautiful and most ideal instrument. I prefer
an a cappella vocal sound that is perfectly tuned vertically to just
about any sound on earth (and possibly in heaven, chuckle).

When the choral sound is perfectly tuned, there is a powerfully deep,
rich, buzzing sensation produced by the difference tones that is
incredibly, powerfully satisfying and completely lacking in
Setharized music. This is not to detract from his accomplishments or
the value of his work. I'm just stating facts as I perceive them.
There is something lacking that I get from harmonic timbres in JI
that I would not want to have to do without musically, but I like to
play as much as anyone, so it sure has its applications and I
wouldn't want anyone to think I feel that I'm "above" using his
techniques.

--- In tuning@y..., BobWendell@t... wrote:

> Bob now:
> Well, neither would be a center frequency in the case I analyzed
> (square wave modulation of a center frequency at the arithmetic
mean.)

But we do hear each of them as a _pitch_ (perhaps with some error)
when the modulation is slow enough. That's the case I'm talking about
(or as close to it as I can get in this "modulation" framework).

> I'm
> not mathematically competent to analyze the more complex case of
> stepped modulation that would have no definable center frequency.

Well then, I'm afraid I don't buy your "intuitive" arguments.

> Bob now:
> It seems to be relevant in the simple case of the trill. Again,
> that's why I reduced the problem to this simple case. The
fundamental
> principle affecting determinacy should not change.

But we _don't_ hear the pitch of a trill as the center, do we?

> Why? I think this simple case is perfectly extensible to the
stepped
> modulation representing a scale or arpeggio. What difference could
it
> make in principle that we went back to the starting pitch instead
of
> to a third pitch? The point is that nothing audibly different is
> introduced by this modulation scheme until we reach a modulating
> frequency high enough to start generating musically spurious
> frequencies that have nothing to do with the two original tones
> between which we were switching, so if we don't hear that, we're
> doing ok on our determinacy.

*sigh*
But if each segment of steady-state pitch is only of finite duration,
shouldn't the classical UP apply to _each_ of them? Except this time,
the bandwidth won't necessarily be symmetrical about the given pitch?

> Precisely, but this is never a practical problem in normal
> performance, is it? If you deliberately introduce it in an electric
> composition, then you wanted it, right? so where' the problem?

It's not a problem, but it's not what I'm talking about, either.

> Empirally speaking, we can perceive extremely high rates of
> arpeggiation of just chords and recognize their purity or lack of
it
> without a problem.

I don't find that to be true at all. Shall we set up a test?

--- In tuning@y..., BobWendell@t... wrote:
>
> Paul, I heard the "Truth on a Bus" file at Sethares' site long ago.
> I'd like to hear more plain, unSetharized 19-tET to see how much I
> like it compared to 12-tET with normal harmonic timbres, though.

Yes, an A/B/C/D comparison would be good here.
>
> I find Sethares' work fascinating. I called up my gifted violinist
> sister and got her to listen to "Turqoise Dabo Girl" in 10-tET.

That's 11-tET! I'm trying to download it from

http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html

but the download keeps getting stuck.

> She
> flipped out. Said the "Star Wars bar scene" should have used that
> music instead of what they did use. I truthfully would not have
> predicted that such tunings could sound as consonant as they do by
> simply playing iwth harmonic content.

Well then, what of all those arguments we've had here where you
maintained that eliminating clashing between partials had nothing to
do with acheiving excellent harmony, while I claimed that it's often
as important a factor as either of the others that were mentioned.
Perhaps it's not so simple as you've made it out to be?

> However, in practice I have to deal with the human voice, and there
> we have no slack on such partial manipulation. I happen to think
the
> human voice is the most beautiful and most ideal instrument. I
prefer
> an a cappella vocal sound that is perfectly tuned vertically to
just
> about any sound on earth (and possibly in heaven, chuckle).
>
> When the choral sound is perfectly tuned, there is a powerfully
deep,
> rich, buzzing sensation produced by the difference tones

and perhaps other phenomena

> that is
> incredibly, powerfully satisfying and completely lacking in
> Setharized music.

Agreed. A couple of years ago, I coined the term "periodicity buzz"
for this phenomenon, in reference to some of Joe Monzo's music: see
<http://www.ixpres.com/interval/monzo/haircut/haircut.htm>.

Note that with harmonic timbres, Utonalities only exploit Sethares-
consonance, which Otonalities exploit that _as well as_ "periodicity
buzz"; comparing the two, particularly beyond the 5-limit, reveals
that otonalities are more consonant than utonalities (with enough
room for other chord types to fall in-between).

> This is not to detract from his accomplishments or
> the value of his work. I'm just stating facts as I perceive them.

As am I . . .

> There is something lacking that I get from harmonic timbres in JI
> that I would not want to have to do without musically,

Could it be, though, that the mapping of partials to 19-tET might not
be a large enough change to destroy the "periodicity buzz" effect? I
suggest we conduct some listening experiments to decide (Bill might
be of great help).

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
>
> Well then, what of all those arguments we've had here where you
> maintained that eliminating clashing between partials had nothing
to
> do with acheiving excellent harmony, while I claimed that it's
often
> as important a factor as either of the others that were mentioned.
> Perhaps it's not so simple as you've made it out to be?
>

Bob:
I claimed that eliminating clashing between partials has nothing to
do with consonant harmony?!?!?!?! I said it wasn't the only, and in
the case of short durations, not necessarily the most important
criterion. Even the combinatorial tones, whatever their source(s)
(neurological, non-linearities, etc.), are reinforced by differences
in the partials of the two tones.

The rich, deep buzz comes when the combinatorial tones and THEIR
HARMONICS line up cleanly with the fundamentals and partials of the
original physical tones that generate them. I even quoted, if I
recall correctly, from an appendix in my pending choral workshop that
explicitly refers to this alignment among all these elements of
consonance, and referred to the pitch at which this alignment
manifests as a "sweet spot".

I wrote all of this well before I joined this list, so I'm not
switching positions on you at all. I think you have way overstated
any position I ever took in this.

Paul:
> Could it be, though, that the mapping of partials to 19-tET might
not
> be a large enough change to destroy the "periodicity buzz" effect?
I
> suggest we conduct some listening experiments to decide (Bill might
> be of great help).

Quite possibly not. That would be interesting to investigate. My
experience as I recall it was quite satisfying in the classical JI
sense when I heard "Truth on a Bus". I have it on my hard drive as I
do the 11-tET (sorry) "Turquoise Dabo Girl". I'll give the former a
listen again and see if I'm agreeing with myself (chuckle). Hope so.
It's nice to be agreed with, even if you have to do it yourself
sometimes.

--- In tuning@y..., BobWendell@t... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> >
> >
> > Well then, what of all those arguments we've had here where you
> > maintained that eliminating clashing between partials had nothing
> to
> > do with acheiving excellent harmony, while I claimed that it's
> often
> > as important a factor as either of the others that were
mentioned.
> > Perhaps it's not so simple as you've made it out to be?
> >
>
> Bob:
> I claimed that eliminating clashing between partials has nothing to
> do with consonant harmony?!?!?!?!

That's how I read it.
>
> I wrote all of this well before I joined this list, so I'm not
> switching positions on you at all. I think you have way overstated
> any position I ever took in this.

Sorry I misinterpreted you -- it was an honest miscommunication, and
I hope you forgive me. Should we care to dig in the archives, I'm
sure I could show you exactly how this impression was given off --
but hopefully we can leave this be.
>
> Paul:
> > Could it be, though, that the mapping of partials to 19-tET might
> not
> > be a large enough change to destroy the "periodicity buzz"
effect?
> I
> > suggest we conduct some listening experiments to decide (Bill
might
> > be of great help).
>
> Quite possibly not. That would be interesting to investigate. My
> experience as I recall it was quite satisfying in the classical JI
> sense when I heard "Truth on a Bus". I have it on my hard drive as
I
> do the 11-tET (sorry) "Turquoise Dabo Girl".

Can you e-mail those to me? For some reason the downloading just
keeps getting stuck for me. What's the tuning/timbre on "Truth on a
Bus"?

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

/tuning/topicId_29265.html#29593

>
> Actually, Gene is talking about a different, but related, notion.
> Rather than a multidimensional space where each point represents a
> different pitch, Gene is referring to a multidimensional space
where each point represents a different precise tuning system. To
begin to
> understand what is involved here, one might want to look at Herman
> Miller's plot of ETs on a 2-dimensional plane, with one axis
> representing the error in the major third, and the other axis
> representing the error in the fifth. What you see is that the
points
> form straight lines, criss-crossing the graph in many directions.
One
> of the lines connects all the meantone ETs (26, 19, 50, 31, 43, 55,
> 12); one of the lines connects all the schismic ETs; one of the
lines connects all the diaschismic ETs, etc.

Hmmm... thanks, Paul! Sounds like that might make a good Monzo web
graphic... Monz??

___________ ________ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_29265.html#29593
>
> >
> > Actually, Gene is talking about a different, but related, notion.
> > Rather than a multidimensional space where each point represents
a
> > different pitch, Gene is referring to a multidimensional space
> where each point represents a different precise tuning system. To
> begin to
> > understand what is involved here, one might want to look at
Herman
> > Miller's plot of ETs on a 2-dimensional plane, with one axis
> > representing the error in the major third, and the other axis
> > representing the error in the fifth. What you see is that the
> points
> > form straight lines, criss-crossing the graph in many directions.
> One
> > of the lines connects all the meantone ETs (26, 19, 50, 31, 43,
55,
> > 12); one of the lines connects all the schismic ETs; one of the
> lines connects all the diaschismic ETs, etc.
>
> Hmmm... thanks, Paul! Sounds like that might make a good Monzo web
> graphic... Monz??
>
What's wrong with Herman Miller's original? Oh, I'll tell you what's
wrong -- it's gone! Fortunately, his description of it still exists:
read /tuning-math/message/390

Bob:
> > Empirally speaking, we can perceive extremely high rates of
> > arpeggiation of just chords and recognize their purity or lack of
> it
> > without a problem.
>
Paul:
> I don't find that to be true at all. Shall we set up a test?

Bob: Good! I'm not sure how correct my statement is until we test it.
I would be glad to know either way, right or wrong. Of course, we
still wouldn't know, though, whether any possible inability to
distinguish was a result of intrinsic indeterminacy or simply a
limitation in human perception, but at least we would know where the
limits lie, whatever the reason.

By the way, a lot of the dialog in your post to which this is a
response has me answering to bits of totally unrelated material. I
don't see how anyone else reading it would be able to figure out what
I was responding to, since what immediately preceeds my remarks
sometimes has nothing to do with what follows.

On the symmetry vs. asymmetry issue, I don't think I've succeeded in
making myself clear yet, and I take at the very least a great deal of
the responsiblity for that. I don't have the tools to deal with it
more rigorously and the argument I'm attempting to use doesn't seem
to be getting through unblemished.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> Sorry I misinterpreted you -- it was an honest miscommunication,
and
> I hope you forgive me. Should we care to dig in the archives, I'm
> sure I could show you exactly how this impression was given off --
> but hopefully we can leave this be.
> >
Chuckle. Nothing to forgive.

I can see how you might get that impression from reading certain
statements in isolation, but taken in the broader context of what I
was saying at the time, I think you will find support for the
position you now perceive me as having. Not taking certain statements
in that broader context is something I've addressed before.

Having said that, if I've never done anything worse than that in my
life, I'll let you buy me a Wal-Mart halo for Christmas, Paul!

Bob:
I have it on my hard drive as
> I
> > do the 11-tET (sorry) "Turquoise Dabo Girl".
>
Pual:
> Can you e-mail those to me? For some reason the downloading just
> keeps getting stuck for me. What's the tuning/timbre on "Truth on a
> Bus"?

Bob:
Gladly, my friend!

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> To begin to
> > understand what is involved here, one might want to look at
Herman
> > Miller's plot of ETs on a 2-dimensional plane, with one axis
> > representing the error in the major third, and the other axis
> > representing the error in the fifth.

> Hmmm... thanks, Paul! Sounds like that might make a good Monzo web
> graphic... Monz??

For the purposes of visualization, one can even put intevals and ets
into the same space, so that lattice points in it represent both
p-limit intervals and p-limit ets. Thus (12, 19, 28) represents the
12-et in the 5-limit, and (19, 30, 44) the 19-et. The plane defined
by (0, 0, 0), (12, 19, 28) and (19, 30, 44) also passes through other
lattice points, representing meantone ets--(31, 49, 72),
(50, 79, 116), (55, 87, 128), etc. The plane itself is perpendicular
to the line running from (0, 0, 0) to (-4, 4, -1), which represents
the diatonic comma of 2^(-4) 3^4 5^(-1) = 81/80.

The line from 0 to the 22-et is *not* perpedicular to this line,
meaning 22 is not a meantone system. It does define a space with the
12-et which is perpendicular to the line to (11, -4, -2),
representing the diaschisma of 2048/2025; other ets lying in the
plane perpendicular to the diaschisma are 10, 14, 34, 46, 58 and 68.
Because the diaschisma is farther from the origin that the diatonic
comma, we might regard 22 in the 5-limit as less close to the 12-et
than the 19 or 31 ets are; however, it *is* close to 12 in the
7-limit.

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

/tuning/topicId_29265.html#29622

> > I find Sethares' work fascinating. I called up my gifted
violinist sister and got her to listen to "Turqoise Dabo Girl" in 10-
tET.
>
> That's 11-tET! I'm trying to download it from
>
> http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html
>
> but the download keeps getting stuck.
>

This crashed *my* browser as well...

>
> Note that with harmonic timbres, Utonalities only exploit Sethares-
> consonance, which Otonalities exploit that _as well
as_ "periodicity
> buzz"; comparing the two, particularly beyond the 5-limit, reveals
> that otonalities are more consonant than utonalities (with enough
> room for other chord types to fall in-between).
>

This is related, I believe, to some of the listening experiments that
Paul set up on the "Tuning Lab"...(near the top of the page) where
utonalities tended to have a peculiar "warble" that otonalities did
not...

The comparison was between the JI 1/7:1/6:1/5:1/4 and the JI 5:6:7:9

http://artists.mp3s.com/artists/140/tuning_lab.html

___________ ________ _________
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> This is related, I believe, to some of the listening experiments
that
> Paul set up on the "Tuning Lab"...(near the top of the page) where
> utonalities tended to have a peculiar "warble" that otonalities did
> not...
>
> The comparison was between the JI 1/7:1/6:1/5:1/4 and the JI 5:6:7:9
>
>
> http://artists.mp3s.com/artists/140/tuning_lab.html

The other chords lower down in the page are relevant as well.

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

/tuning/topicId_29265.html#29632

> >
> What's wrong with Herman Miller's original? Oh, I'll tell you
what's wrong -- it's gone! Fortunately, his description of it still
exists:
> read /tuning-math/message/390

I wish he would put this back up... since I never saw this one...

_______ _____ _______
Joseph Pehrson

--- In tuning@y..., BobWendell@t... wrote:

/tuning/topicId_29265.html#29633

> By the way, a lot of the dialog in your post to which this is a
> response has me answering to bits of totally unrelated material. I
> don't see how anyone else reading it would be able to figure out
what
> I was responding to, since what immediately preceeds my remarks
> sometimes has nothing to do with what follows.
>

I think it might be a good idea for some of us to label some of the
dialogues, as has been suggested.

Just out of curiosity... is it not true that when the rate of pitch
presentation increases... let's say in the case of faster
arpeggiation, the accuracy of the tuning assessment of the listener
decreases??

_________ ________ __________
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_29265.html#29632
>
> > >
> > What's wrong with Herman Miller's original? Oh, I'll tell you
> what's wrong -- it's gone! Fortunately, his description of it still
> exists:
> > read /tuning-math/message/390
>
> I wish he would put this back up... since I never saw this one...

Here's a cooler version I just made:

/tuning/files/perlich/equaltemp.jpg

Here you can clearly see the mistuning each ET has in all three of
the consonant 5-limit interval classes. This also gives a better
sense of how distant each of the ETs are from JI and from one another.

I'll replicate Herman's message, adapted for this graph:::

The column all the way on the right -- 15, 42, 27, 39, 12 -- these
are ETs where the diesis (128:125) vanishes. Thus they all have major
thirds of 400 cents, and are equally distant from the just major
third line.

Connecting 7 on the far left, to 12, with a straight line, one passes
through all the meantone ETs: 26, 45, 88, 69, 50, 81, 31, 74, 43, 55.
These are the ETs where the syntonic comma (81:80) vanishes.

Slightly above that line, just below (and parallel to) the just
perfect fifth line, we see all the multiples of 12: 48, 60, 72, 96,
and of course 12. There are the ETs where the Pythagorean comma
vanishes. They all have fifths of 700 cents: thus they are equally
distant from the just perfect fifth line.

Locating 22 in the upper-left "sextant", and connecting it to 12 with
a straight line, one passes through all the diaschismic ETs: 22, 78,
56, 90, 34, 80, 46, 58, 70, 12. There are the ETs where the
diaschisma (2048:2025) vanishes. This implies that the 45:32 is
represented by exactly 1/2 octave, or 600 cents.

Connecting 25 at the top, to 16 at the bottom, one passes through 25,
22, 63, 41, 60, 79, 19, 35, and 16. These are the ETs where 3125:3072
vanishes. This is related to Graham's MAGIC set of temperaments --
I'll let him explain.

Coneecting 29 (towards the upper left) to 12, one passes through 29,
41, 94, 53, 65, 77, 89, and 12. There are the schismic ETs, where the
schisma (32805:32768) vanishes. As you can see, they all have very
good fifths.

Who can identify the unison vectors for these lines:

Connecting 15 with 23, one passes through 15, 49, 83, 34, 87, 53, 72,
91, 19, 23.

Connecting 25 to 12 one passes through 25, 37, 49, 61, 73, and 12.

What others can you find?

I wrote,

> Connecting 7 on the far left, to 12, with a straight line, one
passes
> through all the meantone ETs: 26, 45, 88, 69, 50, 81, 31, 74, 43,
55.

19 of course is there too, between 45 and 88, and almost on top of
the just minor third line.

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

> Who can identify the unison vectors for these lines:

> Connecting 15 with 23, one passes through 15, 49, 83, 34, 87, 53,
72,
> 91, 19, 23.

This is 34+19, so I presume it must be a minor third.

> Connecting 25 to 12 one passes through 25, 37, 49, 61, 73, and 12.

25+12 I don't recall ever thinking about, but from 25/12~2 I get
2/25 < 3/37 < 1/12, with a semitone, more or less, as generator. One
might use sqrt(2) rather than 2 as the interval of equivalence with
this generator, in which case you have the 50 and 74 ets; that's
interesting because they are also both meantones.

> What others can you find?

Two points determine a line, so pick any two ets on your chart and
you have a line--for instance 41+31 giving you the Miracle line. The
31+22 line, through 53, 75, 84 and 97, is clearly visible, which
makes me happy as I'm still working away on my Orwell piece. We can
see two generators each for the 118 and 171 ets, from 65+53 and 84+34
for 118, and 87+84 and 99+72 for 171.

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

/tuning/topicId_29265.html#29650

> Here's a cooler version I just made:
>
> /tuning/files/perlich/equaltemp.jpg
>
> Here you can clearly see the mistuning each ET has in all three of
> the consonant 5-limit interval classes. This also gives a better
> sense of how distant each of the ETs are from JI and from one
another.
>
> I'll replicate Herman's message, adapted for this graph:::
>

This is great... I don't believe I've ever seen anything quite like
this before...

________ _______ ________
Joseph Pehrson

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Who can identify the unison vectors for these lines:
>
> > Connecting 15 with 23, one passes through 15, 49, 83, 34, 87, 53,
> 72,
> > 91, 19, 23.
>
> This is 34+19, so I presume it must be a minor third.

No, the unison vector is not a minor third (you don't get full credit
unless you read the question carefully!) :)

> > Connecting 25 to 12 one passes through 25, 37, 49, 61, 73, and 12.
>
> 25+12 I don't recall ever thinking about, but from 25/12~2 I get
> 2/25 < 3/37 < 1/12, with a semitone, more or less, as generator.

And the UV?

> One
> might use sqrt(2) rather than 2 as the interval of equivalence with
> this generator, in which case you have the 50 and 74 ets; that's
> interesting because they are also both meantones.

Hmm . . . sqrt(2) doesn't sound like any sort of equivalence to
me . . . but perhaps I misunderstand.

> > What others can you find?
>
> The
> 31+22 line, through 53, 75, 84 and 97, is clearly visible, which
> makes me happy as I'm still working away on my Orwell piece.

And the 5-limit UV is . . . ?

> We can
> see two generators each for the 118 and 171 ets, from 65+53 and
84+34
> for 118, and 87+84 and 99+72 for 171.

Neat -- I only went up to 99 to keep it from getting too
crowded . . . I didn't realize that 7 fell off the end, though :(

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> No, the unison vector is not a minor third (you don't get full
credit
> unless you read the question carefully!) :)

Sorry, teach, it's the kleisma, as in kleismic temperament.

> > > Connecting 25 to 12 one passes through 25, 37, 49, 61, 73, and
12.
> >
> > 25+12 I don't recall ever thinking about, but from 25/12~2 I get
> > 2/25 < 3/37 < 1/12, with a semitone, more or less, as generator.

> And the UV?

I'm going to have to calculate these, I see. This one is
262144/253125, which I can't find on Manuel's web page.

> > One
> > might use sqrt(2) rather than 2 as the interval of equivalence
with
> > this generator, in which case you have the 50 and 74 ets; that's
> > interesting because they are also both meantones.
>
> Hmm . . . sqrt(2) doesn't sound like any sort of equivalence to
> me . . . but perhaps I misunderstand.

It's a 1/2 octave.

> > > What others can you find?
> >
> > The
> > 31+22 line, through 53, 75, 84 and 97, is clearly visible, which
> > makes me happy as I'm still working away on my Orwell piece.
>
> And the 5-limit UV is . . . ?

3^3*5^7/2^21

> > We can
> > see two generators each for the 118 and 171 ets, from 65+53 and
> 84+34
> > for 118, and 87+84 and 99+72 for 171.

> Neat -- I only went up to 99 to keep it from getting too
> crowded . . . I didn't realize that 7 fell off the end, though :(

65+53 is the apotheosis of schismic temperament, and this line is
just the schisma line, but the others are not familiar to me: 84+34
gives us 84+34 gives 2^23 3^6 5^(-14), 87+84 gives
2^(-44) 3^(-3) 5^(21), and 99+72 I find especcially interesting--it
is 2 3^(-27) 5^(18) = 2 (25/27)^9. Did you know nine large limmas was
so close to an octave?

--- In tuning@y..., genewardsmith@j... wrote:

> I'm going to have to calculate these, I see. This one is
> 262144/253125, which I can't find on Manuel's web page.
>
> > > One
> > > might use sqrt(2) rather than 2 as the interval of equivalence
> with
> > > this generator, in which case you have the 50 and 74 ets; that's
> > > interesting because they are also both meantones.
> >
> > Hmm . . . sqrt(2) doesn't sound like any sort of equivalence to
> > me . . . but perhaps I misunderstand.
>
> It's a 1/2 octave.

Well, I knew that . . . (?)
>
> > > > What others can you find?
> > >
> > > The
> > > 31+22 line, through 53, 75, 84 and 97, is clearly visible, which
> > > makes me happy as I'm still working away on my Orwell piece.
> >
> > And the 5-limit UV is . . . ?
>
> 3^3*5^7/2^21

I see that this temperament is #5 in Graham's 5-limit rankings, in addition to the #3 ranking that
this system gets in the 11-limit.

> 99+72 I find especcially interesting--it
> is 2 3^(-27) 5^(18) = 2 (25/27)^9. Did you know nine large limmas was
> so close to an octave?

I did not know that! Wild, weird stuff, Gene!

On Fri, 26 Oct 2001 19:34:12 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>What's wrong with Herman Miller's original? Oh, I'll tell you what's
>wrong -- it's gone! Fortunately, his description of it still exists:
>read /tuning-math/message/390

Hmm? It's still there.

http://www.io.com/~hmiller/png/et-scales.png

Ah, but I remember that IOCOM was going to be down for a bit while they
were moving their servers to a new location. You must have tried to visit
at just the wrong time. It seems to be working now.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning@y..., jpehrson@r... wrote:
>
> This is great... I don't believe I've ever seen anything quite like
> this before...

I made a zoomed-out version:

/tuning/files/perlich/equaltemp2.jpg

In this one, 7-, 8-, 9-, 10-, and 18-tET become visible (they were
off the map before). Also, you can immediately see the _direction_ as
well as the magnitude of the error of each of the consonant
intervals . . . I only went up to 72 this time to prevent
overcrowding (though 69 and 50 stil overlap on the meantone
line . . .)

On Mon, 29 Oct 2001 22:18:37 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>--- In tuning@y..., jpehrson@r... wrote:
>>
>> This is great... I don't believe I've ever seen anything quite like
>> this before...
>
>I made a zoomed-out version:
>
>/tuning/files/perlich/equaltemp2.jpg
>
>In this one, 7-, 8-, 9-, 10-, and 18-tET become visible (they were
>off the map before). Also, you can immediately see the _direction_ as
>well as the magnitude of the error of each of the consonant
>intervals . . . I only went up to 72 this time to prevent
>overcrowding (though 69 and 50 stil overlap on the meantone
>line . . .)

This is great! Just one problem, though: where's 21? It ought to be between
9 and 12.

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--- In tuning@y..., Herman Miller <hmiller@I...> wrote:

> This is great! Just one problem, though: where's 21? It ought to be
between
> 9 and 12.

You guessed it -- consistency. If I included 21, I'd have to put it
in two places, corresponding to

major third = 7 (best)
minor third = 6 (best)
perfect fifth = 13

and

major third = 7 (best)
minor third = 5
perfect fifth = 12 (best)

If you retort that the major third is more consonant than the minor
third, I'll come right back at you saying the major sixth is more
consonant than the minor sixth . . .

My least favorite Blackwood etudes are the ones in 21 and 22. yuk.

On Tue, 30 Oct 2001 02:46:41 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>--- In tuning@y..., Herman Miller <hmiller@I...> wrote:
>
>> This is great! Just one problem, though: where's 21? It ought to be
>between
>> 9 and 12.
>
>
>You guessed it -- consistency. If I included 21, I'd have to put it
>in two places, corresponding to

Ah, right. I get it. Forgot about 21 being inconsistent (although barely
so: both "minor thirds" are almost equally as bad).

>If you retort that the major third is more consonant than the minor
>third, I'll come right back at you saying the major sixth is more
>consonant than the minor sixth . . .

Or one might argue that 21-TET doesn't have an actual minor third: 5 is
closer to 7/6 and 6 is closer to 11/9.

>My least favorite Blackwood etudes are the ones in 21 and 22. yuk.

21 is really not well suited for diatonic music; it just sounds "out of
tune" when used that way. But it has a unique "shadowy" character that
could potentially have some interesting uses, as I discovered when retuning
Galticeran (the 21-ET version is my current favorite, even above the 12-ET
original).

http://www.io.com/~hmiller/midi/Galticeran21.mid

--
languages of Azir------> ---<http://www.io.com/~hmiller/lang/index.html>---
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

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

/tuning/topicId_29265.html#29733

> --- In tuning@y..., jpehrson@r... wrote:
> >
> > This is great... I don't believe I've ever seen anything quite
like
> > this before...
>
> I made a zoomed-out version:
>
> /tuning/files/perlich/equaltemp2.jpg
>
> In this one, 7-, 8-, 9-, 10-, and 18-tET become visible (they were
> off the map before). Also, you can immediately see the _direction_
as
> well as the magnitude of the error of each of the consonant
> intervals . . . I only went up to 72 this time to prevent
> overcrowding (though 69 and 50 stil overlap on the meantone
> line . . .)

This really made it a lot clearer what was going on with the "narrow-
wide" business... Thanks!

________ _______ ________
Joseph Pehrson

This is truly revealing, Paul! Thank you! Very useful stuff for
making the "big picture" easily accessible.

Gratefully,

Bob

> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_29265.html#29650
>
> > Here's a cooler version I just made:
> >
> > /tuning/files/perlich/equaltemp.jpg
> >

--- In tuning@y..., Herman Miller <hmiller@I...> wrote:

> 21 is really not well suited for diatonic music; it just
sounds "out of
> tune" when used that way. But it has a unique "shadowy" character
that
> could potentially have some interesting uses, as I discovered when
retuning
> Galticeran (the 21-ET version is my current favorite, even above
the 12-ET
> original).
>
> http://www.io.com/~hmiller/midi/Galticeran21.mid

Wow . . . there are a lot of interesting harmonies here -- some which
sound "just", others moderately to extremely dissonant -- also some
very non-diatonic melodic scales in a few places . . . can you talk
about any of this?

On Wed, 31 Oct 2001 01:01:07 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>Wow . . . there are a lot of interesting harmonies here -- some which
>sound "just", others moderately to extremely dissonant -- also some
>very non-diatonic melodic scales in a few places . . . can you talk
>about any of this?

It's based on a non-diatonic scale that I described in a message back on
June 3.

D# A#=Bb 7 19
B F# 0 12 (21-TET version)
D A 5 17
A#=Bb F 19 10

The first section introduces the scale in its original form, then modulates
to a transposed version of the scale. Near the end, I use two different
transpositions of the scale simultaneously for some really "crunchy" (I
believe the technical term is), dissonant harmonies. If I have some time
tomorrow, I'll scan the piano score (the 12-ET original, written in 1986)
and put it up on my web page. The 21-ET version benefits from the good 7/4
approximation (only 2.6 cents sharp) of the minor seventh.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

On Wed, 31 Oct 2001 00:04:17 -0500, Herman Miller <hmiller@IO.COM> wrote:

>If I have some time
>tomorrow, I'll scan the piano score (the 12-ET original, written in 1986)
>and put it up on my web page. The 21-ET version benefits from the good 7/4
>approximation (only 2.6 cents sharp) of the minor seventh.

And here it is, in four pages.

http://www.io.com/~hmiller/png/Galticeran-p1.png
http://www.io.com/~hmiller/png/Galticeran-p2.png
http://www.io.com/~hmiller/png/Galticeran-p3.png
http://www.io.com/~hmiller/png/Galticeran-p4.png

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin