octaves and partials

Greetings,
I dunno,
There were so many tag lines preceding it, I lost track, but nevertheless,
It was posted:

>> "2:4" is only one possibility. other possibilities
>> include "1:2", "3:6", "4:8", etc. all are slightly different
>> intervals, but all are wider than 1200 cents.

The need to make sense of cents(applicable to struck piano strings).
The "octave" determines the width of the ET semitones, and "octave"
covers a broad range of widths. Since a cent is 1/100th of an ET semitone,
can we go about defining the octave as 1200 of them? Simple, piece of cake,
except for the fact that an octave can be perfect in various sizes. An equal
temperament can be tuned within a 2:1 or a 6:3 octave, but both of them will be
1200 cents when you have all the other notes equally spaced between them.
A wider beginning temperament octave will steepen the curves on either
side of the middle three octaves of the piano. It will have smoother fifths
and faster thirds than a more compressed octave.

Ed Foote

--- In tuning@yahoogroups.com, a440a@a... wrote:
> Greetings,
> I dunno,
> There were so many tag lines preceding it, I lost track, but
nevertheless,
> It was posted:
>
> >> "2:4" is only one possibility. other possibilities
> >> include "1:2", "3:6", "4:8", etc. all are slightly different
> >> intervals, but all are wider than 1200 cents.
>
> The need to make sense of cents(applicable to struck piano strings).
> The "octave" determines the width of the ET semitones,
and "octave"
> covers a broad range of widths. Since a cent is 1/100th of an ET
semitone,
> can we go about defining the octave as 1200 of them? Simple, piece
of cake,
> except for the fact that an octave can be perfect in various
sizes. An equal
> temperament can be tuned within a 2:1 or a 6:3 octave, but both of
them will be
> 1200 cents when you have all the other notes equally spaced between
them.

this goes against the standard definition of cents. 1200 cents is a
frequency ratio of 2, it is not a variable measure which adjusts
according to how you tune your piano. any tuning machine will measure
cents in this way and not in the variable way you're suggesting.

on 6/6/03 2:05 PM, wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:

> --- In tuning@yahoogroups.com, a440a@a... wrote:
>> Greetings,
>> I dunno,
>> There were so many tag lines preceding it, I lost track, but
> nevertheless,
>> It was posted:
>>
>>>> "2:4" is only one possibility. other possibilities
>>>> include "1:2", "3:6", "4:8", etc. all are slightly different
>>>> intervals, but all are wider than 1200 cents.
>>
>> The need to make sense of cents(applicable to struck piano strings).
>> The "octave" determines the width of the ET semitones,
> and "octave"
>> covers a broad range of widths. Since a cent is 1/100th of an ET
> semitone,
>> can we go about defining the octave as 1200 of them? Simple, piece
> of cake,
>> except for the fact that an octave can be perfect in various
> sizes. An equal
>> temperament can be tuned within a 2:1 or a 6:3 octave, but both of
> them will be
>> 1200 cents when you have all the other notes equally spaced between
> them.
>
> this goes against the standard definition of cents. 1200 cents is a
> frequency ratio of 2, it is not a variable measure which adjusts
> according to how you tune your piano. any tuning machine will measure
> cents in this way and not in the variable way you're suggesting.

It seems to me that maybe what was meant by the now unknown person that you
responded to is that for the given partial being used to determine
zero-beat, you have a frequency ratio of 2, and therefore 1200 cents
regardless.

In any case the whole question makes me realize there is something a little
vague about tuning numbers being applied when the partials used for
determining zero-beat will vary from octave to octave. The article on
perceived pitch (which I think was your reference yesterday) makes me think
that one way to interpret tuning numbers applied in a stretched situation is
to relate the numbers to the perceived pitch. But it would be a lot of work
to try to apply that theory to a practical situation. On the other hand the
tuning numbers probably don't apply to the fundamental.

How can some sense be made of tuning cents on an instrument with
inharmonicity? It seems to me the only "real" tuning numbers are the
unstretched one's you sit down with when trying to tune an
instrument--unless you have some model of the instrument that lets you do
better than that. In other words when the tuning is all done on a stretched
instrument you really can't give tuning number specs for the actual
resulting tuning in any obvious way that I can see.

-Kurt Bigler

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

> It seems to me that maybe what was meant by the now unknown person
that you
> responded to is that for the given partial being used to determine
> zero-beat, you have a frequency ratio of 2, and therefore 1200
cents
> regardless.

for the given partial being used, you have a frequency ratio of 1,
or 0 cents.

> The article on
> perceived pitch (which I think was your reference yesterday) makes
me think
> that one way to interpret tuning numbers applied in a stretched
situation is
> to relate the numbers to the perceived pitch.

too ambiguous, will vary from ear to ear. what's done is to relate
them to the *fundamentals*. it's a sensible standard.

> On the other hand the
> tuning numbers probably don't apply to the fundamental.

which ones?

> How can some sense be made of tuning cents on an instrument with
> inharmonicity?

professional piano tuning devices focus in on the fundamentals, and
have built-in stretch tuning presets that one can target.

> It seems to me the only "real" tuning numbers are the
> unstretched one's you sit down with when trying to tune an
> instrument--unless you have some model of the instrument that lets
you do
> better than that.

why wouldn't you?

> In other words when the tuning is all done on a stretched
> instrument you really can't give tuning number specs for the actual
> resulting tuning in any obvious way that I can see.

it's done all the time. goes by the fundamentals. just a convenient
standard.

>How can some sense be made of tuning cents on an instrument with
>inharmonicity? It seems to me the only "real" tuning numbers are the
>unstretched one's you sit down with when trying to tune an
>instrument--unless you have some model of the instrument that lets you
>do better than that. In other words when the tuning is all done on a
>stretched instrument you really can't give tuning number specs for the
>actual resulting tuning in any obvious way that I can see.

It's true. There are models that measure the inharmonicity of the
sound, and have a fixed set of rules that adjusts the tuning accordingly.
Many 'pro' portable electronic piano tuners do this, or claim to.

But with a bearing plan, there's really no need to do any of this.
It is the choices of the human ear which the automated models only
claim to approximate. In this area, which I've called "timbre country",
human judgement defines the correct tuning, the bearing plan is only
a... plan. Different people may judge differently. Pianists sometimes
get very fussy about their tuner, whose tunings they like.

-Carl

on 6/7/03 12:58 AM, wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> It seems to me that maybe what was meant by the now unknown person
> that you
>> responded to is that for the given partial being used to determine
>> zero-beat, you have a frequency ratio of 2, and therefore 1200
> cents
>> regardless.
>
> for the given partial being used, you have a frequency ratio of 1,
> or 0 cents.

Ah, of course. And yet that doesn't reveal much about the stretch. I guess
I was thinking you would tune zero beat (for example on the 6th partial of
the lower note and the 3rd partial of the octave) and then do something like
compare (in the same example) the 3rd partial of both notes to determine a
"meaningful" stretch for that octave, in cents. The stretch of the
fundamental just seems less meaningful, especially if it is dropping out of
the pitch-perception process in some cases.

>> The article on
>> perceived pitch (which I think was your reference yesterday) makes
> me think
>> that one way to interpret tuning numbers applied in a stretched
> situation is
>> to relate the numbers to the perceived pitch.
>
> too ambiguous, will vary from ear to ear. what's done is to relate
> them to the *fundamentals*. it's a sensible standard.
>
>> On the other hand the
>> tuning numbers probably don't apply to the fundamental.
>
> which ones?

The scale specification.

>> How can some sense be made of tuning cents on an instrument with
>> inharmonicity?
>
> professional piano tuning devices focus in on the fundamentals, and
> have built-in stretch tuning presets that one can target.

It makes sense in that case, when you have a mechanical device that measures
fundamentals. In a sense that seems the only justification for using the
fundamental, but a very good one.

>> It seems to me the only "real" tuning numbers are the
>> unstretched one's you sit down with when trying to tune an
>> instrument--unless you have some model of the instrument that lets
> you do
>> better than that.
>
> why wouldn't you?

I really meant a good model of the perceived pitch. That's what you
probably wouldn't have, we agree on that.

>> In other words when the tuning is all done on a stretched
>> instrument you really can't give tuning number specs for the actual
>> resulting tuning in any obvious way that I can see.
>
> it's done all the time. goes by the fundamentals. just a convenient
> standard.

The fundamental would seem to be the only possible choice, if in fact
anything works at all. Based on discussions earlier about how the
fundamental really is "not it" for the low piano notes, I just figured the
fundamental didn't make sense either.

But if you had to chose something arbitrarily I guess the fundamental would
have to be it. More standard, if less meaningful. It still strikes me that
a theory of stretch measurement based on comparing partials as I described
above might be in some way useful, or at least more insightful as to how
stretch actually functioned. This is just a hunch, and I am not married to
the idea.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 6/7/03 12:58 AM, wallyesterpaulrus <wallyesterpaulrus@y...>

> >> On the other hand the
> >> tuning numbers probably don't apply to the fundamental.
> >
> > which ones?
>
> The scale specification.

normally that does apply to the fundamental, even with extremely
inharmonic timbres such as those of indonesian gamelan. what did you
have in mind?

> >> How can some sense be made of tuning cents on an instrument with
> >> inharmonicity?
> >
> > professional piano tuning devices focus in on the fundamentals,
and
> > have built-in stretch tuning presets that one can target.
>
> It makes sense in that case, when you have a mechanical device that
measures
> fundamentals. In a sense that seems the only justification for
using the
> fundamental, but a very good one.

it's also the most convenient standard. totally unambiguous.

> >> In other words when the tuning is all done on a stretched
> >> instrument you really can't give tuning number specs for the
actual
> >> resulting tuning in any obvious way that I can see.
> >
> > it's done all the time. goes by the fundamentals. just a
convenient
> > standard.
>
> The fundamental would seem to be the only possible choice, if in
fact
> anything works at all. Based on discussions earlier about how the
> fundamental really is "not it" for the low piano notes, I just
figured the
> fundamental didn't make sense either.
>
> But if you had to chose something arbitrarily I guess the
fundamental would
> have to be it. More standard, if less meaningful. It still
strikes me that
> a theory of stretch measurement based on comparing partials as I
described
> above might be in some way useful, or at least more insightful as
to how
> stretch actually functioned.

yes, but there are simply too many unknown in this calculation that
would affect the result -- loudness, musical context, subjective
octave stretch, etc. even the left ear and right ear of the same
subject show different relationships between frequency and perceived
pitch.

on 6/7/03 2:02 PM, wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 6/7/03 12:58 AM, wallyesterpaulrus <wallyesterpaulrus@y...>
>
>>>> On the other hand the
>>>> tuning numbers probably don't apply to the fundamental.
>>>
>>> which ones?
>>
>> The scale specification.
>
> normally that does apply to the fundamental, even with extremely
> inharmonic timbres such as those of indonesian gamelan. what did you
> have in mind?

At risk of repeating myself, I guess what I would say is that a scale
specification makes sense only as an idealized thing, needing interpretation
by a tuner on a strongly inharmonic instrument. Any attempt for an
after-the-fact tuning spec for such an instrument strikes me as having very
limited meaning UNLESS it refers to some theory of perception allowing some
kind of _likely_ perceived pitch to be computed, and that will probably not
be based on the fundamental, or unless it specifies the pitch of the
"important" partials.

I was under the impression that there _is_ no "fundamental" present on some
bells, unless you call the lowest partial the fundamental in spite of it's
clearly _not_ representing the perceived pitch, or unless you start talking
about an implied fundamental, which clearly depends on a theory, or else on
data from listening trials.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 6/7/03 2:02 PM, wallyesterpaulrus <wallyesterpaulrus@y...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >> on 6/7/03 12:58 AM, wallyesterpaulrus <wallyesterpaulrus@y...>
> >
> >>>> On the other hand the
> >>>> tuning numbers probably don't apply to the fundamental.
> >>>
> >>> which ones?
> >>
> >> The scale specification.
> >
> > normally that does apply to the fundamental, even with extremely
> > inharmonic timbres such as those of indonesian gamelan. what did
you
> > have in mind?
>
> At risk of repeating myself, I guess what I would say is that a
scale
> specification makes sense only as an idealized thing, needing
interpretation
> by a tuner on a strongly inharmonic instrument. Any attempt for an
> after-the-fact tuning spec for such an instrument strikes me as
having very
> limited meaning UNLESS it refers to some theory of perception
allowing some
> kind of _likely_ perceived pitch to be computed, and that will
probably not
> be based on the fundamental, or unless it specifies the pitch of the
> "important" partials.

but if all the tones have the same, or even approximately the same,
deviations from harmonicity, then all the perceived pitches are simply
shifted pretty much a uniform amount relative to the fundamentals.
this amount doesn't affect any of the intervals of the tuning, so the
tuning still "makes sense".

> I was under the impression that there _is_ no "fundamental" present
on some
> bells, unless you call the lowest partial the fundamental in spite
of it's
> clearly _not_ representing the perceived pitch, or unless you start
talking
> about an implied fundamental, which clearly depends on a theory, or
else on
> data from listening trials.

right. bells are so inharmonic they really don't sound like single
pitches, but rather multiple pitches. still, the whole pattern of
objective and subjective pitches pretty much transposes itself
unchanged as one goes to higher and lower bells. while the harmonic
series isn't simply "stretched", it's still almost identical from bell
to bell. so a tuning system that "makes sense" for bells can pretty
much be implemented just by tuning any given fixed partial to it. i'm
sure there's such a standard for bell tuning already, which you can
probably look up fairly easily.

Posted:

>> >> How can some sense be made of tuning cents on an instrument with
>> inharmonicity?
> professional piano tuning devices focus in on the fundamentals,
> and > have built-in stretch tuning presets that one can target.
>> It makes sense in that case, when you have a mechanical device that
> measures fundamentals. In a sense that seems the only justification for
> using the fundamental, but a very good one.
>it's also the most convenient standard. totally unambiguous.

Greetings,
The state of the art tuning machines don't use the fundamental, at all.
Pitches are identified by their particular partials, and the art of tuning a
piano is knowing when to measure what partial, either by ear or machine.
The lower in pitch, the less fundamental there is, as we go up, there are
less and less partials in the spectrum. The machines focus on the partials
because the fundamentals are less important, from the standpoint of control,
than the partials, which beat a lot faster when tempering. Within
non-perceptible changes in the fundamentals, a tuner can alter partials to create a wide
variety of tempering,(an endeavor of some attraction to the afflicted few).
Also, the piano's partials are not all perfectly consistant. You will
often notice small numbers on the piano plates, under the strings. The top
notes will be 13, then perhaps six notes later 13.5, then 14, and so on. These
are diameter changes, made in an attempt to keep the tension progressing evenly
higher as the strings get longer and the pitch gets lower. This pattern of
wire sizes and string lengths is known as the "scaling" of that piano. It
takes into account the impedance of the soundboard and size of piano to attempt a
balanced, even, response in loudness and sustain, while still producing a
musical set of intervals.
The tension in a string determines the amount of inharmonicity, and when
you have 6 or 8 notes in a row with the same size wire, you will have a change
in tension,(thus inharmonicity) whereever they another size wire. At the
point where it changes sizes, the highest tension first note will be next to the
lowest tension last note of the preceding wire size. This makes the partials
land all over the chart. The tuner's job is to compromise things enough. We
have to work among the partials to do this, and when doing so, the fundamental
is rarely the prime consideration.
Regards,
Ed Foote
www.uk-piano.org/edfoote/
www.uk-piano.org/edfoote/well_tempered_piano.html
<A HREF="http://artists.mp3s.com/artists/399/six_degrees_of_tonality.html">
MP3.com: Six Degrees of Tonality</A>

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> while the harmonic series isn't simply "stretched", it's still
> almost identical from bell to bell.

Ahem, are you *sure* on this one? Or are you just talking out of your hat?

Bells are cast metal objects. Allowing for one or two foundries that *maybe* have advanced and refined the casting process to a complete science, and duplicity is not really a problem, I'd say that it would be difficult to come up with objects that have a greater chance of variation from item to item than bells.

With variations in metal content, wall thickness, difference in the 'machining' of the bells (by hand or machine) after the casting process, etc., I would be surprised if you could find a carillon or bellplay that you could do a harmonic analysis and across the board find a real and significant similarity in the harmonic spectrum of each and every individual bell.

But this only conjecture based on playing many metal objects over the years, and having visited a number of carillons. I'm just curious where you get the info to make the above statement.

Cheers,
Jon

Hello Szanto Jon!

I think you are right about most bells, but there are a few
specialists (english, I think?) who make carillon bells designed to
have several tones by striking in different, particular places on the
bell. But this is extreme precision work -- rocket science, you might
say.

I was helping my teacher this weekend to tune up an iron bonang
(using a ballhead hammer and piece of railroad track), and did a
series of FFTs with matlab, thinking this might help. The smallest
change in the shape of the kettles changed the spectrum. Prediction?
Impossible. I quote Dr. Wolf: "Idiophones are idiosyncratic."

Szia!

Gabor

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > while the harmonic series isn't simply "stretched", it's still
> > almost identical from bell to bell.
>
> Ahem, are you *sure* on this one? Or are you just talking out of
your hat?
>
> Bells are cast metal objects. Allowing for one or two foundries
that *maybe* have advanced and refined the casting process to a
complete science, and duplicity is not really a problem, I'd say that
it would be difficult to come up with objects that have a greater
chance of variation from item to item than bells.
>
> With variations in metal content, wall thickness, difference in
the 'machining' of the bells (by hand or machine) after the casting
process, etc., I would be surprised if you could find a carillon or
bellplay that you could do a harmonic analysis and across the board
find a real and significant similarity in the harmonic spectrum of
each and every individual bell.
>
> But this only conjecture based on playing many metal objects over
the years, and having visited a number of carillons. I'm just curious
where you get the info to make the above statement.
>
> Cheers,
> Jon

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > while the harmonic series isn't simply "stretched", it's still
> > almost identical from bell to bell.
>
> Ahem, are you *sure* on this one? Or are you just talking out of
your hat?
>
> Bells are cast metal objects. Allowing for one or two foundries
that *maybe* have advanced and refined the casting process to a
complete science, and duplicity is not really a problem, I'd say that
it would be difficult to come up with objects that have a greater
chance of variation from item to item than bells.
>
> With variations in metal content, wall thickness, difference in the
'machining' of the bells (by hand or machine) after the casting
process, etc., I would be surprised if you could find a carillon or
bellplay that you could do a harmonic analysis and across the board
find a real and significant similarity in the harmonic spectrum of
each and every individual bell.
>
> But this only conjecture based on playing many metal objects over
the years, and having visited a number of carillons. I'm just curious
where you get the info to make the above statement.
>
> Cheers,
> Jon

i'm totally confident of my statement. there was an article on bells
that was quite a heavy subject of discussion here a while back. the
good quality carillon bells are all made to conform to a fixed
inharmonic spectrum, featuring in one point a minor third instead of
a major third over the fundamental, among other significant
departures from harmonicity. the casting technique involved, and the
particular choice of spectrum, were developed over many centuries,
just like brass instruments, etc . . . totally empirically. you can
hear that a scale played on a good carrillon will sound like a
particular "chord" being transposed uniformly upward. at least the
carillons i've lived around.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> i'm totally confident of my statement.

Why does that not surprise me? :)

I guess as long as it carries the parts of the sentence "most" bells, and you are speaking about the contemporary carillons of the highest quality. From visiting a large number of cathedrals and churches and other places one would find bells (mostly in Germany, England, and a few in France), I heard an amazing array of spectra in the various bells. And I guess part of the difference would be the categories of 'bell tower' bells and actual carillons (bells cast for musical purposes).

Cheers,
Jon

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

/tuning/topicId_44236.html#44319

> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > i'm totally confident of my statement.
>
> Why does that not surprise me? :)
>
> I guess as long as it carries the parts of the sentence "most"
bells, and you are speaking about the contemporary carillons of the
highest quality. From visiting a large number of cathedrals and
churches and other places one would find bells (mostly in Germany,
England, and a few in France), I heard an amazing array of spectra in
the various bells. And I guess part of the difference would be the
categories of 'bell tower' bells and actual carillons (bells cast for
musical purposes).
>
> Cheers,
> Jon

***To digress just slightly, but I think maybe I'll be forgiven since
this is such a funny story. My *favorite* bell of all time is the
one sitting inside the Kremlin. There is a huge bell there, maybe 15
feet high, just sitting on the ground. It was the engineering
miscalculation of all time. The bell was too heavy to put in the
cathedral structure and somehow they found out too late, so it just
sits there, and they cut a side out of it so that tourists could go
inside there...

JP