TUNING digest 1392

Regarding the post about the Sala Five Improvisations- it actually is available
on CD but very hard to find. I saw it at Rasputin's in Berkeley but they woul
dn't sell it to me as it was not yet priced. I asked them to price it and hold
it for me and when I returned to get it they had sold it!!!!

Allen Strange

> You'd be better off pointing out that few of the acoustic instruments
> with exact harmonic partials (or rather, the players of such
> instruments) can produce a pitch which does not waver more than 1 cent.
>
If this is the case then why are there so many complaints about synthesizer
1 cent tuning accuracy when acoustic musicians can't do any better?

John Loffink
jloffink@pdq.net

On Thu, 23 Apr 1998, Loffink, John wrote:
> > You'd be better off pointing out that few of the acoustic instruments
> > with exact harmonic partials (or rather, the players of such
> > instruments) can produce a pitch which does not waver more than 1 cent.
>
> If this is the case then why are there so many complaints about synthesizer
> 1 cent tuning accuracy when acoustic musicians can't do any better?

Because pianos, organs, and harpsichords can do _much_ better.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

>Bowed strings could be a prominent exception, but I doubt if anybody
>could convincingly make that case for most any other orchestral instrument.
>It's almost trivial to demonstrate that that's not true of winds in
>general: All you have to so is play them onto an oscilloscope; in many
>cases, you can quite clearly see the higher harmonics "walking" through the
>oscillogram.

I'm afraid I have to disqualify the oscilloscope here. The oscilloscope
will have a certain finite sampling time and the period of the waveform,
varying slighlty as it does, will never be precisely matched to this
sampling time. The oscilloscope utilizes a certain approximation of
Fourier's theorem, not the theorem itself. If you study the mechanics of
a 1-component driver, such as a bow, reed, or lips, you will understand
that regardless of the extent to which the resonant modes of vibration
of the instrument deviate from a harmonic series, the driving mechanism
will force the waveform to become periodic, which implies exact integer
overtones. Don't confuse the resonant modes of vibration with the
spectrum of the sound (the two are closely related but not identical!)

Looping involves the same timing problems as oscilloscope sampling.

J. Kukula had more to say on this subject, is he around?

> . . . JI does what it does more in the lower harmonics
>than the upper ones

I think Harry Partch would disagree with you there. To the extent that
higher harmonics are musically relevant at all, the associated intervals
require more precise tuning in order to engage those harmonics than do
the intervals associated with lower harmonics. On the other hand, it is
true that the difference in sensory consonance is greater when you
detune a simple ratio by a small amount than when you detune a complex
ratio by a small amount. But if you are using more complex ratios as
consonances, their sensory consonance is only just barely enough to
justify that usage, so you'd better be careful how you tune them! (A
simplification, no doubt, but I thought I had to make the point
concisely here).

>Last time we had this discussion, we agreed (I thought) that bowed
>strings, winds, and brass instruments had exact integer harmonics for as
>long as they sustain a consistent tone.

Bowed strings could be a prominent exception, but I doubt if anybody
could convincingly make that case for most any other orchestral instrument.
It's almost trivial to demonstrate that that's not true of winds in
general: All you have to so is play them onto an oscilloscope; in many
cases, you can quite clearly see the higher harmonics "walking" through the
oscillogram.

Now bowed strings, since they have very exact nodal points (at least in
the cases of bridges and frets) that may not be true. The nodal point at
the reed end of a typical wind pipe is inherently much more elastic in
nature though. (Air-pressure node that is; air-velocity antinode.)

But in the case of the comparatively imprecise nodal-point that a
finger-pad provides in unfretted strings, I wouldn't be surprised at all if
different overtones vary in exactly where they conclude the node is. I
don't recall any specifics in this regard, but could investigate.

Also, having worked a lot with sampled tones looped on a single
vibration, which obviously renders them exactly harmonic, I can certainly
attest to the fact that the timbre changes slightly when it drops into
looping. And that timbre changes in a manner consistent with overtone
detuning.

Now bear in mind that I'm talking about very small deviations from exact
harmonics here - probably, just as an estimate, on the order of 2-4 cents
or so. So when we start demanding tuning accuracy on the order of tenths
of a single cent, they start becoming significant.

Again though, as I aluded in my earlier message, these deviations from
exact harmonics are much smaller in the lower harmonics than in the higher
ones. To the degree that JI does what it does more in the lower harmonics
than the upper ones, perhaps that makes this consideration a small one.

> From: david first
>
> > > So what is the minimum pitch resolution that is acceptable? Is this
> for
> > > music comprised mostly of long sustained harmonies?
>
> Well - this is actually a tricky question. I make my samples in CPS, not
> cents
> so if one is starting with a base freq of 440 then multiplying by 3/2,
> 5/4...27/16....3645/2048...etc., the amount of decimal places obviously
> varies.
>
> Ultimately, you have to use your ears. If you hear beating/phasing between
> a
> given pair of frequencies, then something's off.
>
Could you give me a rough figure in terms of Hertz/CPS or even beats per
minute/hour/day/year/millenium? :-) I realize the issue with beating, but
not everyone doing just intonation feels you need 100% beatless harmonies,
especially if one is doing band/orchestral type music. This question is
open to all list members using synthesizers/samplers: how much pitch
resolution does your own music need?

> To answer your second question, yes, I am talking - at least in my case -
> about sustained tones. But I suppose that it goes, for me, beyond what one
> can
> get "away with" regarding tuning. It is true that tuning errors are more
> obvious in "slower" music, but isn't that the crux of the issue? Play ANY
> music fast enough and I suppose the errors become negligible. 12et was,
> and
> is, consider close enough to JI for most people. I presume the the main
> reason
> for this forum is to explore the alternatives to the alternatives
> regarding
> pitch and to not accept what is given as gospel whether tuning systems or
> hardware. Perhaps I was a bit contentious in my original post, but I was
> hoping to find out how others feel about this particular issue, and more
> importantly, if and how anyone is, in fact, going beyond what the
> synthesizer
> manufacturers are saying is good enough.
>
Synthesizer manufacturers say 1 cent resolution is good enough because:
1. They don't understand the concepts of just intonation very well and...
2. No one has ever quantified what tuning resolution is good enough. I've
heard many times that 1 cent resolution is poor, but I've never heard in
terms of cents or Hertz what _was sufficient. If .003 Hz is what it takes
to make someone happy, then I suggest a system like Kyma, as mass market
manufacturers are unlikely to implement so fine a resolution. However, if
0.1 or 0.05 Hz is good enough for microtonalists and current technology is
0.5 Hz resolution, then synthesizer manufacturers may be willing to improve
their systems to that extent and we have something to work with.

> From: Paul Hahn
>
> > If this is the case then why are there so many complaints about
> synthesizer
> > 1 cent tuning accuracy when acoustic musicians can't do any better?
>
> Because pianos, organs, and harpsichords can do _much_ better.
>
What is the best resolution one can expect, either in terms of cents or
Hertz?

> From: mr88cet@texas.net (Gary Morrison)
>
> >If this is the case then why are there so many complaints about
> synthesizer
> >1 cent tuning accuracy when acoustic musicians can't do any better?
>
> Well, we're clearly asking for more than you can realistically expect
> of
> a human performer (on an indefinite-pitched instrument that is). In my
> view anyway, that's perfectly fine; that's one of the benefits of
> technology
>
Good answer!

> From: mr88cet@texas.net (Gary Morrison)
>
> >I'm afraid I have to disqualify the oscilloscope here. The oscilloscope
> >will have a certain finite sampling time and the period of the waveform,
> >varying slighlty as it does, will never be precisely matched to this
> >sampling time.
>
> I don't know what you're getting at by introducing sampling into this,
> but you certainly don't need a digitizing oscilloscope to see this. A
> plain, boring ol' analog scope will show it just fine.
>
> If you zoom it in on a single period of the tone, you see the peaks and
> troughs of the vibration bobbing up and down, and sometimes moving around
> too. Regardless of what you attribute that to, it's certainly not a
> strictly periodic wave.

This is the varying phase shift of the individual frequencies of the tone.
You are right, it has nothing to do with sampling as it's visible on an
analog oscilloscope. You could also interpret these phase shifts as
microscopic shifts in frequencies of the harmonics.

John Loffink
jloffink@pdq.net

>If this is the case then why are there so many complaints about synthesizer
>1 cent tuning accuracy when acoustic musicians can't do any better?

Well, we're clearly asking for more than you can realistically expect of
a human performer (on an indefinite-pitched instrument that is). In my
view anyway, that's perfectly fine; that's one of the benefits of
technology.

Just to make sure that this discussion is not getting too far off the
original topic, let me paraphrase my original suggestion: If you go to a
synthesizer manufacturer asking for tuning resolution on the order of a few
tenths of a single cent, it would probably be wise to ask yourself whether
the tones you're realizing with it are even that close to being periodic.

Now Paul here is calling into question my belief that typical orchestral
instruments have deviations from purely harmonic vibrations. These
deviations, best I can tell, are extremely slight, but almost certainly
much more than a few tenths of a cent. I have not, however, specifically
measured them; I have only observed them to exist.

Now back to your regularly-scheduled discussion.




>I'm afraid I have to disqualify the oscilloscope here. The oscilloscope
>will have a certain finite sampling time and the period of the waveform,
>varying slighlty as it does, will never be precisely matched to this
>sampling time.

I don't know what you're getting at by introducing sampling into this,
but you certainly don't need a digitizing oscilloscope to see this. A
plain, boring ol' analog scope will show it just fine.

If you zoom it in on a single period of the tone, you see the peaks and
troughs of the vibration bobbing up and down, and sometimes moving around
too. Regardless of what you attribute that to, it's certainly not a
strictly periodic wave. But if you change the time per division so as to
zoom out to show 10 or so cycles, you can see that the bobbing up and down
of the peaks and troughs follows a regular progression from one cycle to
the next. Sometimes you can even actually see a peak moving through the
lower frequency components.

This, by the way occurs on fundamental-mode pitches as well as
"overblown" pitches, so it can't be due to a subharmonic.





>If you study the mechanics of
>a 1-component driver, such as a bow, reed, or lips, you will understand
>that regardless of the extent to which the resonant modes of vibration
>of the instrument deviate from a harmonic series, the driving mechanism
>will force the waveform to become periodic,

Would you apply that to a flute, for example?

It sounds like you're suggesting that lips, or a reed for example, can
only vibrate periodically. I personally am aware of no evidence to that
effect. They certainly don't have the physical characteristics of a system
that vibrates only in harmonics, notably a mechanical transmission line.





>> . . . JI does what it does more in the lower harmonics
>>than the upper ones
>I think Harry Partch would disagree with you there.

That of course is why I said "to the degree that JI does what it
does...": To include the possibility that that degree might be zero. Or
very high.

In other words, I was not expressing an opinion along those lines; just
stating that if that were true, then the fact that the lower harmonics are
closer to harmonic than the would be relevant.

Gary,

>If you zoom it in on a single period of the tone, you see the peaks and
>troughs of the vibration bobbing up and down, and sometimes moving around
>too. Regardless of what you attribute that to, it's certainly not a
>strictly periodic wave. But if you change the time per division so as to
>zoom out to show 10 or so cycles, you can see that the bobbing up and >down
>of the peaks and troughs follows a regular progression from one cycle to
>the next. Sometimes you can even actually see a peak moving through the
>lower frequency components.

I grant you that typically the wave will not be strictly periodic,
because the player's bowing, lipping, or blowing will not be entirely
steady.

The problem is that if one of the harmonic partials is changing in
amplitude due to this unsteadiness, and you do a Fourier analysis on a
segment of time in which these changes are significant, the Fourier
transform will only be allowed to assign one amplitude to each
frequency. Therefore, the result of the analysis will be that the
frequencies are slightly altered, usually both upwards and downwards
simultaneously, so that interference can simulate the changes in
amplitude. (Try it!) However, as the ear can only resolve one pitch
within each critical bandwidth (and there are nerve-firing periodicities
which supplement the cochlear mechanisms), it will indeed be able to
interpret the signal as having harmonic partials of changing amplitudes.
I'm not saying that one interpretation or the other is more correct; the
problem of describing a waveform as a sum of frequencies each of whose
amplitude changes over time has many equally valid solutions. I'm saying
that the solution relevant for how we hear the tones and how dissonance
might arise with other tones is, in most cases, one where the partials
are harmonic and of changing amplitude. (Looking at peaks and troughs
can be misleading. The way we hear a wave has surprisingly little
correlation with how the wave looks.)

Add the variablity of the overall pitch level and you have another
complication for a traditional Fourier analysis.

If there are indeed overtones of constant amplitude that do not fall
within the harmonic series of the fundamental, there must be an
independent source of energy producing them. I don't think this occurs
too often in any significant proportion in the types of instruments
we're talking about.

>>If you study the mechanics of
>>a 1-component driver, such as a bow, reed, or lips, you will understand
>>that regardless of the extent to which the resonant modes of vibration
>>of the instrument deviate from a harmonic series, the driving mechanism
>>will force the waveform to become periodic,

>Would you apply that to a flute, for example?

I don't think the turbulent airflow that drives a flute is well enough
understood to make this characterization. However, this airflow does
appear to be characterised by "periodic" and "chaotic" regimes of
behavior.

>It sounds like you're suggesting that lips, or a reed for example, can
>only vibrate periodically. I personally am aware of no evidence to that
>effect. They certainly don't have the physical characteristics of a system
>that vibrates only in harmonics, notably a mechanical transmission line.

Under a steady airflow, the Bernoulli effect that explains their motion
predicts a periodic behavior. Even without the rest of the instrument to
dictate where the fundamental will resonate with a minimum of input
energy, the mouthpieces of these instruments can be used to create a
musical tone with harmonic overtones (albeit highly variable overall
pitch). The timbre of instruments, as I'm sure you know, is highly
dependent on the ability of the instrument to resonate at harmonic
overtones of a given resonance. This is becuase the lips/reed are highly
non-linear, creating a host of harmonic overtones along with the
sinusoudal fundamental that a simple Bernoulli model would predict. Now
the extent of this non-linearity is dependent on the physical
characteristics of the reed/lips, which can be constantly altered by the
player, even if he/she is not meaning to do so. Hence the amplitude of
the partials can vary, sometimes wildly, rendering the waveform
aperiodic. However, for the time domains relevant for the ear's
analysis, this does not lead to any relevant alterations in the
frequencies of the partials.

>>> . . . JI does what it does more in the lower harmonics
>>>than the upper ones
>>I think Harry Partch would disagree with you there.

>That of course is why I said "to the degree that JI does what it
>does...": To include the possibility that that degree might be zero. Or
>very high.

>In other words, I was not expressing an opinion along those lines; just
>stating that if that were true, then the fact that the lower harmonics are
>closer to harmonic than the would be relevant.

>Boy, am I confused!

>I grant you that typically the wave will not be strictly periodic,
>because the player's bowing, lipping, or blowing will not be entirely
>steady.

If so, I would have expected it to be audible, and also moderately
random. The variations I'm referring to are generally neither.




>If there are indeed overtones of constant amplitude that do not fall
>within the harmonic series of the fundamental, there must be an
>independent source of energy producing them. I don't think this occurs
>too often in any significant proportion in the types of instruments
>we're talking about.

Other than noise elements of course, but that's a different topic of course.




>Under a steady airflow, the Bernoulli effect that explains their motion
>predicts a periodic behavior...

I'll presume that you've read more on the topic than I have, although
I've certainly read a fair amount.




>>...if that were true, then the fact that the lower harmonics are
>>closer to harmonic than the would be relevant.
>>Boy, am I confused!

Forget it; It was a tautology: If it's true that JI properties are
more related to lower-harmonics, then the tuning of the lower harmonics is
more important than that of the higher harmonics.

The point though is that the lower harmonics seem to be much more
periodic than the higher ones, since the overall waveshape (which is
dictated by the lower harmonics) seems to stay pretty consistent, but the
details (dictated by the higher harmonics) vary a lot more.

> The point though is that the lower harmonics seem to be much more periodic
> than the higher ones, since the overall waveshape (which is dictated by the
> lower harmonics) seems to stay pretty consistent, but the details (dictated
> by the higher harmonics) vary a lot more.

Finishing that thought: So if indeed it is true that JI does what it
does mostly in the lower harmonics, then the apparent fact that the higher
harmonics are not quite exactly harmonically tuned, is probably not as
interesting. That would make this concern I'm raising less of a concern.

But again, Paul has described theoretical reasons to think that that
"apparent fact" is an illusion in, or misinterpretation of, the
oscillograms. If indeed he's right, then the concern I raised is not a
concern at all.

On Fri, 24 Apr 1998, Loffink, John wrote:
> From: Paul Hahn
> > [ . . . ] pianos, organs, and harpsichords can do _much_ better
> > [than 1 cent resolution].
>
> What is the best resolution one can expect, either in terms of cents or
> Hertz?

I don't know what the best one can expect is, but I seem to recall piano
tuners on the list mentioning that their students are expected to
achieve 0.1 cent accuracy on their piano tuning exams, and that seems
about the right ballpark to me in terms of what I would like from a
synthesizer. 1-cent resolution isn't really accurate enough to capture
the subtle differences between various shades of meantone, as someone
else (I forget who) pointed out over the weekend.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

>Brass instruments really are aperiodic. I read about this since
>the last time it came up. The worst tuned partials are the low
>ones for a conical tube. Even valve horns rarely play the
>fundamental for this reason.
> Graham Breed

This is another case of confusing the modes of resonance with the
spectrum of an actual tone. The former is not a harmonic series but
depends on the shape of the instrument. Good brass makers sacrifice the
first resonance so that the others can come close to a harmonic series.
Still, the lowest resonances are typically dozens of cents off a
harmonic series. But the tone itself contains exactly harmonic partials.
If you don't believe me, read Dave HIll's recent post, where he found
that brass partials are a fraction of a cent off of harmonic, which is
far less than the dozen cent type errors in the lowest modes of
resonance. Even if you don't believe that Dave Hill's methodology led to
the errors of a fraction of a cent, clearly the modes of resonance and
the spectrum of a played tone are clearly two different things.

The reason the "fundamental," or more precisely the lowest resonant
mode, isn't used by horn players is that none of its harmonic overtones
coincide with resonances of the instrument. Hence mode-locking can't
occur, and a steady tone can't be produced, let alone one with a nice
timbre. A different, quiet "fundamental" can, however, be played by a
skilled lipper, which is of the "subharmonic" variety: The harmonic
overtones are engaging the resonances of the instrument, keeping the
lips vibrating in a steady pattern, while the fundamental is far from
the lowest resonance and is only weakly amplified at best.

Ken Wauchope wrote,

>Some time back I did some audio experiments to determine the extent to
>which I could actually discriminate beating in just intervals.
>Listening to rather loud synthesized sawtooth waves through headphones
>at about 440Hz,

That's amazing! I did the exact same experiment many years ago. I swept
through all the intervals within an octave, and wrote down all the
points at which beating stopped.

>I found that the limit of my discrimination was at the
>19 limit -- for example I could manage to tune a 19/13 by eliminating
>beats, but not a 21/13.

Very close to my results! The most complex ratio I stopped at was 17/13.

Here's something I posted on New Year's Eve. The people advocating
really exact JI really need to think about this:

For an interval in perfect just intontation, whatever phase difference
was present at the beginning of the sound would persist for the duration
of the sound. Let's say two instruments are producing tones, both of
which have an equally loud partial at 1000Hz. For someone standing in a
particular location with respect to the two instruments, there is some
delay time between the beginning times of the two instruments for which
the 1000Hz components will interefere constructively, resulting in four
times the energy at that frequency than what one instrument alone would
produce. For a delay time just 1/2000 of a second longer, you get
destructive interference, or no energy at that frequency. Obviously no
human instrumentalist can control their onset time to within 1/2000 of a
second relative to the onset time of another instrument. So you end up
with a random number between zero and four to describe the energy of the
1000Hz component -- probably not the musical effect you were looking
for. Even an individual performer playing both instruments can't do it
-- try tuning two synth keys to the same note and notice how repeatedly
playing both keys "simultaneously" leads to random fluctuations in the
loudness.

Now if the onset times are controlled really accurately by MIDI, and
let's say a 90-degree phase shift is chosen so that the energy at 1000Hz
is twice that from one instrument, then you're doing okay, right? But
for someone standing in a different location in the room, though, the
differece in the path lengths from the two instruments, and thus the
phase, will be different. There are nodes of destructive interference
and antinodes of constructive interference in the room no matter what
the original phase shift was. So some members of the audience may be
receiving no energy at 1000Hz, while others may be receiving four times
what they would from one instrument. Again, probably not the effect you
were looking for.

If both instruments are electronic, and their signals are mixed into one
speaker, there will be no spatial nodes or antinodes. So finally you can
control the musical effect. But other than monophonic electronic
MIDI-sequenced music, perfect JI can lead to big problems. In the real
world, thankfully, acoustic instruments are slightly out-of-tune with
each other, so that whatever the onset delay and whatever the position
in the room, one gets a signal that frequencies common to both tones
have an average energy twice that of one instrument. There will not be
noticeable, let alone startling, differences in the musical effect
depending on minute variations in the onset delay or position in the
room.

>

I wrote,

>Now if the onset times are controlled really accurately by MIDI, and
>let's say a 90-degree phase shift is chosen so that the energy at 1000Hz
>is twice that from one instrument, then you're doing okay, right?

>If both instruments are electronic, and their signals are mixed into one
>speaker, there will be no spatial nodes or antinodes.

Note, however, that if the two tones have the same relative phases of
their partials, then their 2000Hz harmonics will have a 180-degree phase
shift and potentially cancel each other out. This is another probably
undesirable musical effect. Add more tones to the chord and the
potential for cancellations increases. Thus I must conclude that under
no circumstances is exact JI desirable, unless all tones are played
through one loudspeaker and one exerts full control of the phases of all
partials of all tones so as to eliminate anomalous interference. (Tones
sound quite similar, but not exactly the same, when the relative phases
of the partials are messed with; the characteristic "brassiness" of
brass instruments is due in some part to the sharp spikes in the
waveform which Gary mentioned.)