Yahoo Tuning Groups Ultimate Backup tuning Piano history and inharmonicty

Well it's been quite some time since I've done a drive-reading, and I
dare say, I'm not so sure if it is comforting or disturbing to see
that some things never seem to change, like the endless debate of
Bach's tuning. I particularly liked Thomas' deconstruction of Eggar's
arguments.

Anyway, re: the historical development of pianos, in particular the
issue of inharmonicity. This is a topic I can speak to with some
authority as it is my primary professional occupation.

Inharmonicty is caused by one thing: the relative rise in significance
as a restoring force of the stiffness of the string over the
normally-operative restoring force of tension. The added force of
stiffness increases the acceleration of the displaced string during
every cycle, and increased acceleration naturally results in decreased
period which raises the frequency of that particular mode of
vibration. It affects higher modes progressively more than lower.
Inharmoncity is ALWAYS present in EVERY vibrating string, it is just
that in many cases it is so low that the difference between the
altered series of overtones and a truly harmonic series is
inconsequential.

In practical terms, stiffness becomes noticeably operative due to
three design factors:

1. Tension. When the ambient tension drops, rigidity becomes
relatively more operative. This is why traditionally keyboard
instruments have been designed to utilize maximum practical tension
levels. I will state that I believe there is no real grounds for the
current low tension fad which some harpsichord makers are peddling.

2. Length/diameter ratio. When diameter becomes larger relative to
length, rigidity increases. Simple tube/beam models.

3. Stiffness of the material. Different metals have different values
of Young's Modulus (E), and the higher the value, the more rigidity,
thus the more inharmonicty, all other factors being equal (tension,
length, diameter, freq.). The Great Divide is the move from
charcoal-fired steel refining to coal-fired, which occurred near the
end of the first half of the 19-century. Charcoal introduces trace
amounts of phosphorous, which is the factor which makes the material a
steel and not pure iron. This was the operative element from antiquity
until 1840 or so. Coal, on the other hand, leaves trace amounts of
carbon, which is the basis of modern steels. The two together produce
an extremely brittle material which is useless as it cannot be formed,
which is why modern steel production processes involves aspects to
control any P contamination.

As of today, nobody has reproduced P wire in commercial quantities.
Stephen Birkett in Canada is coming close, but the process is
complicated and expensive. ALL other "historical" wires are made from
carbon-based steels, even if the seller claims they are "iron". Based
on lab tests of surviving historical strings, P wire seems to have
been surprisingly strong while having a value of E significantly lower
the C steel. This makes it more flexible and also raises internal
damping, both of which give a pure (i.e. more harmonic) though less
bright tone.

Historically, the story was pretty simple. The first pianos more or
less had string lengths similar to harpsichords, but it was quickly
discovered that a thicker string works better when the excitation
mechanism is a hammer. Thicker strings require a shorter scale, not
for tension's sake, but because they will break at lower loads
(tension/area). Contrary to what you will read in many books, it is
not true that all diameters of a given material will break a the same
pitch, a falsehood based on the fact that while tension varies, load
is constant for all diameters at a given length/freq. Steel (and to a
lesser extent, brass) work-hardens when cold formed, and wire drawing
is a type of cold-forming. Thus finer diameters are relatively
stronger than fatter, and when you replace a fine string with a
thicker you will find that it cannot be drawn up to the same pitch
before it breaks.

Thus the early makers where caught in a bind regarding inharmonicty:
thicker strings required shorter scales, and inharmonicty rose because
the were getting a double whammy on the diameter/length ratio: strings
where becoming both shorter and thicker. From 1800 or so onward, the
move to ever thicker strings was further encouraged by a desire for
ever louder instruments, largely due to the gradual demise of the
patronage of royalty and the emergence of a relatively wealthy middle
class which desired to ape the life styles of the aristocrats in every
possible way, which included the "consumption" of cultural events.
Thus the appearance of the mega-concert, by which the artist earns by
packing ever more people into ever larger halls.

The increased thickness matched by reduced scale length continued
until the appearance of true carbon steel. By this time, makers had
also been playing around with metal reinforcement, primarily as a
means of stabilizing tuning under variations of temperature and
humidty, to the point where the two separate developments coincided to
allow a return to longer scales with even larger strings. The metal
frameworks allowed the higher tension levels and the stringer wire
allowed a longer scale, which - theoretically - inharmonicty. However,
the increased stiffness of the stronger material added increased
inharmonicity which could not be entirely accounted for by the
increased length, which meant that makers have never been able to
reproduce the relatively pure sound of an early piano in the larger
modern form.

Modern steel is many many times stiffer than old wire. I have some of
the stuff in my shop and I am always amazed t how utterly rigid it is
compared to all the "historical" wire types I have, including some
real antique strings. I have not measured the inharmoncity of a modern
piano, though it would be plenty easy enough to do and perhaps I'll
get around to it when the current work overload reduces a bit. But I
have had the relatively unpleasant experience of tuning a number of
transitional pianos, those made in the period of 1830-50. They
definitely demonstrate a markedly noticeable degree of inharmonicity.
The idea of using anything but equal on these instruments is a joke,
because a pure interval doesn't exist. Equal hides the inherent
inharmoncity because it keeps everything fluttering more or less
equally. If you try to set anything else, what you get in an
unpleasant mix of rapidly beating "good" intervals with rapidly
fluttering "bad" intervals. The problem is the the "good" intervals
aren't good enough to provide the acoustic sense of repose that
unequal temperaments manifest on harmonic instruments. The whole thing
just ends up sounding out of sorts.

Frankly I have never been convinced by the sound of modern pianos in
historical temperaments. The sound is too far from how these
temperaments sound on historical instruments for my ear, but then I
came from the organ/harpsichord world and the modern piano always has
been and still is a bizarre product of advanced technology. It sounds
like a synthesizer to me, the tone being too loud and too long to be
an acoustic instrument.

Finally, regarding the leather/felt dichotomy. The historical move has
been from harpsichord-like to a muffled, dark, perhaps bell-like tone,
but it was by no means any sort of universal evolution, as many books
would have us believe. The earliest instrument had very thin leather,
of in many cases no leather at all, just bare wood against the
strings. This very bright and transparent tone ideal continued until
1805 or so, when Streicher, one of the last hold-outs, made their big
design change. Most other makers had long moved to leather covered
hammers, though the covering was quite thin. English and French makers
had already been using a considerably thicker application of multiple
layers of leather, and some Viennese makers began imitating this by
about 1810, particularly Streicher. His motivation is without
question, as we have surviving letters between him and Haertel saying
he is trying to imitate the sound of English pianos. Other Viennese
makers continued the late 18th century ideal, keeping to a very thin
covering of 2 or 3 layers only until 1830 or so. Graf, another big
name, used many layers, like Streicher. The move to felt was primarily
a practical consideration; it is considerably less labor intensive to
apply one layer of felt than many layers of leather. Felt is also a
more predictable raw material; leather is terrible, the acoustic
quality of skins varies from fantastic to absolutely crap, something
which cannot always be determined before the application. Felt does
have the added advantage that it produces a duller tone, which helped
to hide the inharmonicty. In many ways, the industrial production
aspect of piano making has always been far more the arbiter of design
considerations that the aesthetic, though they moved with a certain
degree of parallelism that makers could claim the latter for
promotional purposes.

Any questions?

Ciao,

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> ....not so sure if it is comforting or disturbing to see
> that some things never seem to change,
> like the endless debate of
> Bach's
alleged
> tuning.
Once in 1999, when I wangeled a dozen 5ths into JSB's squiggles,
in order to fob my old friend H.A.Kellner:

/tuning/topicId_25902.html#25913

then I never expected ~not even in my wildest dreams~
that my own unsmoked hoax would result in so much even
denser smother out of the reeking pipes of:
http://www.taylorandboody.com/opuses/opus_41.htm
"This organ is (de)tuned(?) in an unusual temperament,..."

Apology:
In the hope,
that the squiggle-charlatanry ends up soon fully in smoke,
without any residues.

Please excuse:
Sorry to all those,
that became bored by that entangeled 'Rosetta-stone'd story,
that sometimes got out of my control,
when flogging that dead horse all to much...&.c.t...

> I particularly liked Thomas' deconstruction of
> Eggar's
> arguments.
Me too.
>
> Anyway, re: the historical development of pianos, in particular the
> issue of inharmonicity.
>
> Inharmonicty is caused by one thing: the relative rise in
> significance
> as a restoring force of the stiffness of the string over the
> normally-operative restoring force of tension....
> Inharmoncity is ALWAYS present in EVERY vibrating string,...

Right:
Theoretically it vanishes strictly spoken only for
infintesimal razor-thin and infinite long-expanded strings
in the abstract terms of Fourier's harmonic analysis
as linear approximation.

But in real strings the ratio inbetween length and diameter is finite.
Hence the inevitable bias inbetween the partials,
that causes the observed deviations from JI.

> 2. Length/diameter ratio. When diameter becomes larger relative to
> length, rigidity increases. Simple tube/beam models.
>
....

> Frankly I have never been convinced by the sound of modern pianos in
> historical temperaments.

http://ccrma.stanford.edu/marl/CASL/Files/lottermoser.html
and his assistent Grützmachcher do reccomend for modern pianos
in PC^(1/24) units (~1Cent):

C -4 G -4 D -4 A -4 E -4 B 0 F# 0 C# 0 G# 0 Eb 0 Bb 0 F -4 C

but i do prefer at the moment:

C -1 G -5 D -5 A -5 E -1 B -1 F# -1 C# -1 G# -1 Eb -1 Bb -1 C

when rescaling the inharmonicity of the 5ths
to the half amount than usual applied.

> The sound is too far from how these
> temperaments sound on historical instruments for my ear, but then I
> came from the organ/harpsichord world and the modern piano always
> has
> been and still is a bizarre product of advanced technology.
> It
> sounds
> like a synthesizer to me,
> the tone being too loud.
That depends also on the intonation of the hammers.
http://www.concertpitchpiano.com/WhatsVoicing.html
http://www.pianofundamentals.com/book/en/2.7.1
http://www.pianoworld.com/ubb/cgi-bin/ultimatebb.cgi?ubb=get_topic;f=16;t=000009;p=0

>and too long to be
> an acoustic instrument.
Agreed,
but the slow decay in longer persistent tones
hepls in counting beats when tuning by tremor-rates.

>...the industrial production
> aspect of piano making has always been far more the arbiter of
> design
> considerations that the aesthetic, though they moved with a certain
> degree of parallelism that makers could claim the latter for
> promotional purposes...
...that depends also on the demands of the modern customers.
>
> Any questions?
Yes, in deed:

1.Which kind of piano would satisfy
at soonest yours own personal taste?

2.How far do you allow yourself deviations from ET?

bye
A.S.

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Paul Poletti <paul@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:

on my question:
> > Which kind of piano would satisfy
> > at soonest yours own personal taste?
>
Paul replied:
>....For
> modern monsters, give me a Blüthner or a Bösendorfer any day.

Here my own humble preferences obtained from
playing some pianos at the annual:
http://musik.messefrankfurt.com/frankfurt/en/
by my ~admittetly~ subjective criterion:
How easy can I tame the "monsters" of which trademark?

1.
http://www.bluethner.de/en/unternehmen/uframe.htm
Wilhelm Furtwängler wrote:
"Blüthner instruments can sing,
certainly the best you can say about a piano"
http://en.wikipedia.org/wiki/Bl%C3%BCthner

2.
http://www.ibach.de/eng/ibach-e.htm
"An IBACH instrument is the result of more than 200 years experience
in piano making, manufactured for professional pianists as well as for
an exclusive circle of private customers..."
http://de.wikipedia.org/wiki/Ibach_(Unternehmen)

3.
http://www.boesendorfer.com/index.php?lang=en
"Bösendorfer is among the world's eldest piano manufacturers, rich in
tradition and world famous for the sound and outstanding quality of
its instruments."
http://en.wikipedia.org/wiki/B%C3%B6sendorfer

4.
http://www.bechstein.de/Default.aspx?
"They enchant musicians by their distinctive transparency, the perfect
touch, the balance between dynamic power and a singing, lyric voice."
http://en.wikipedia.org/wiki/C._Bechstein_Pianofortefabrik

5.
http://www.petrof.com/petrof_sounds.html
"....and have very romantic and superb sound."
http://en.wikipedia.org/wiki/Petrof

6.
http://www.fazioli.com/eng/index.php
http://en.wikipedia.org/wiki/Fazioli

7.
http://www.steinway.com/
http://en.wikipedia.org/wiki/Steinway_%26_Sons

8.
http://www.august-foerster.de/
"And thus such inspired designs like the quarter-tone grand piano and
electrochord were introduced in the 1920s and 1930s."
http://www.xs4all.nl/~huygensf/english/wyschnegradsky.html
http://www.xs4all.nl/~huygensf/pics/wysch2.jpg
http://www.piano-lessons-central.com/august-forster-piano.html

9.
http://www.grotrian.de/start.htm
http://de.wikipedia.org/wiki/Grotrian-Steinweg

10.
http://www.schimmel-piano.de/index.php?id=2&L=1
http://en.wikipedia.org/wiki/Wilhelm_Schimmel
http://de.wikipedia.org/wiki/Wilhelm_Schimmel_Pianofortefabrik_GmbH

11.
http://www.kawai.co.jp/english/
http://en.wikipedia.org/wiki/Kawai

12.
http://www.yamaha.co.jp/english/product/piano/
http://en.wikipedia.org/wiki/List_of_Yamaha_products#Acoustic_musical_instruments

13.
http://www.gibson.com/en%2Dus/Divisions/Baldwin/
http://en.wikipedia.org/wiki/Baldwin_Piano

&.c.t.
???
??
?

Paul:
What do you think about ~so called~ modern replicas of the
http://en.wikipedia.org/wiki/Fortepiano
in German the:
http://de.wikipedia.org/wiki/Hammerklavier
( Hersteller von Hammerklavier-Kopien im 20./21. Jahrhundert )
[ Producers of copies in the 20th/21th century ]:

???

* Robert Brown, Oberndorf bei Salzburg
* Ugo Casiglia, Cinisi, Palermo, Italien
* Christoph Kern, Staufen im Breisgau
* Paul McNulty, Divisov, Tschechien
* J. C. Neupert (gegr. 1868), Bamberg
* Andrea Restelli, Milano, Italien
* Michael Walker, Altneudorf bei Heidelberg
* Mirko Weiss, Niederscherli bei Bern, Schweiz
* Ambrosius Pfaff, Locarno, Schweiz

???

http://wiki.hammerfluegel.net/index.php/Hauptseite

bye
A.S.

🔗Paul Poletti <paul@...>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>

> 4.
> http://www.bechstein.de/Default.aspx?
> "They enchant musicians by their distinctive transparency, the perfect
> touch, the balance between dynamic power and a singing, lyric voice."

Yeah, forgot about Bechstein. A very nice piano, or at least was. I
have no idea what they are doing now. Almost ipossible to keep up with
the Chinese crap. Just had tapas with a friend from Berlin a piano
tuner who tells me that Chinese uprights can be had for 900 euro!
That's less than I charge for a rstringing of a 5 octave fortepiano!

>
> Paul:
> What do you think about ~so called~ modern replicas of the
> http://en.wikipedia.org/wiki/Fortepiano
> in German the:
> http://de.wikipedia.org/wiki/Hammerklavier
> ( Hersteller von Hammerklavier-Kopien im 20./21. Jahrhundert )
> [ Producers of copies in the 20th/21th century ]:

Well, you can start be reading the philosopjy pages of my website:

http://www.polettipiano.com

Further information can be had by trolling the archives over at:

/fortepiano/

..but like Hesse's Magic Theatre, it's not for everyone.

>
> * Robert Brown, Oberndorf bei Salzburg

A good friend and a very good builder who gets a sound I can believe in.

> * Ugo Casiglia, Cinisi, Palermo, Italien

Never 'eard of 'im.

> * Christoph Kern, Staufen im Breisgau

Another friend from wayback, builds nice instruments, though his own
design.

> * Paul McNulty, Divisov, Tschechien

I wouldn't know where to start. It's all on the FePo list archives.

> * J. C. Neupert (gegr. 1868), Bamberg

Poor old Neupert. They should give it a rest and retire gracefully.

> * Andrea Restelli, Milano, Italien

Good solid work.

> * Michael Walker, Altneudorf bei Heidelberg

Such a nice guy, I wish his instrument were up to snuff.

> * Mirko Weiss, Niederscherli bei Bern, Schweiz

The name rings a bell but can't say as if I know.

> * Ambrosius Pfaff, Locarno, Schweiz

Never 'eard of 'im.

How any list not include Christopher Clarke, the Father of Us All, as
Beethoven said of Haydn?

Ciao,

🔗Daniel Forró <dan.for@...>

🔗Ozan Yarman <ozanyarman@...>

I do have a Bechstein grand that has a very sharp timbre and whose
treble regions get detuned very quickly when I play fortissimo. So
many strings broke in my inexperienced attempts to tune it. No longer
though, and they have been replaced. The inharmonicity of the strings
are very much noticable whether old or new. The octaves need to be
stretched as much as 2-3 cents. There is a difference of at least one
and a half syntonic commas between the lower C and the one before the
highest.

Oz.

On Nov 29, 2008, at 4:02 AM, Daniel Forró wrote:

> I do regular recitals of historical and jazz music on one new
> Bechstein, middle size Grand. Very nice really. In my opinion sound
> is too sharp and brilliant, one has to be very careful with
> fortissimo, and dynamics and expression generally - it's very
> sensitive and responsing to every nuance. It took me some time to get
> used to it. I prefer Boesendorfer concerning timbre, then Yamaha,
> then Steinway. Steinway has usually too heavy touch, but here in
> Japan they set it down to match muscles of little tender Japanese
> female pianists :-) Which I like (I mean that light touch, not l. t.
> J .f. p.).
>
> Once I met here a historical Bechstein from 70s of 19th century,
> renewed, restored. That one was great.
>
> Daniel Forro
>
> On 29 Nov 2008, at 8:13 AM, Paul Poletti wrote:
>
>> --- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
>> wrote:
>>>
>>
>>> 4.
>>> http://www.bechstein.de/Default.aspx?
>>> "They enchant musicians by their distinctive transparency, the
>> perfect
>>> touch, the balance between dynamic power and a singing, lyric
>> voice."
>>
>> Yeah, forgot about Bechstein. A very nice piano, or at least was. I
>> have no idea what they are doing now. Almost ipossible to keep up
>> with
>> the Chinese crap. Just had tapas with a friend from Berlin a piano
>> tuner who tells me that Chinese uprights can be had for 900 euro!
>> That's less than I charge for a rstringing of a 5 octave fortepiano!
>>
>
>
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🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:

Sorry Paul,

> Yeah, forgot about Bechstein.
Agreed, as far as that concerns to the actual models.

> A very nice piano, or at least was....
... barely once ago.

http://en.wikipedia.org/wiki/C._Bechstein_Pianofortefabrik
"
After the fall of the Berlin Wall

Due to reunification of Germany and elimination of the Berlin Wall,
the land formerly belonging to the Bechstein factory was used for new
construction in the capital....."

Imho:
theirs best instruments were produced
once in the Berlin factory.

>I
> have no idea what they are doing now.
"....However, Karl Schulze, a piano enthusiast and co-owner of
Bechstein, had continued the legacy of fine pianomaking."

He's more sales-man than a piano-maker:
Hence he outsourced the production mainly to China.

" The new Bechstein factories began production of several brand names
under the Bechstein group."

Sadly none of that "group" ever regained the old Berlin quality again.

> "....Hoffmann was the mid-level brand while C....=

They simply sold Hoffmann-pianos, that were
an little bit improved old poor GDA
http://en.wikipedia.org/wiki/East_German
standard pianos with an "Bechstein"-group signboard.

"....Bechstein remained the flagship brand...."
but now even that got manufactored by Hoffmann
under the alleged label 'Bechstein' traditionally.

"....In 1992 Bechstein started a new factory in Saxony with investment
of 15 million Euros...."
...which still then hasn't recovered the old Berlin quality
Imho.

"...In 1996 C. Bechstein went public. By 2006 the company opened eight
upscale showrooms, increasing the number of Bechstein dealerships in
major cities across Europe, North America and Asia...."

Under the slogan: MASS instead of CLASS.

"....Since 2003, Bechstein formed partnership with Samick with the
samick's participation in Bechstein limited to 39%. In 2005 the
Bechstein/Samick joint venture opened their new piano factory in
Shanghai, China...."

http://en.wikipedia.org/wiki/Samick

"Samick also sells Pianos to the German C. Bechstein Pianofortefabrik
who is listed on the Frankfurt Stock Exchange. Major shareholders are
Karl Schulze and Samick (19,5%) of South Korea. [1] In addition,
Bechstein and Samick have a joint venture factory in Shanghai, China."

Hence most todays 'Bechstein's do contain inside 'Samick'.

"... In 2006 a new Bechstein/Hoffman joint factory was opened in
Bohemia, Czech Republic,...."

...so they lost by and by their last
experts from the former Berlin aera.

> Almost ipossible to keep up with
> the Chinese crap.

...now sold under the label 'Bechstein'-group.

> Just had tapas with a friend from Berlin a piano
> tuner who tells me that Chinese uprights can be had for 900 euro!

Far east junk direct direct delivered from the chinese factory,
shipped from Shanghai by 'Bechstein'
to theirs Berlin 'piano-salon'.

> That's less than I charge for a rstringing of a 5 octave fortepiano!

Sadly,
that results in the death sentence for many good old pianos,
that otherwise would be worth to got restringed anew.

Quest:
Stands the old brand "Bechstein" meanwhile for:
Disposable pianos for through-away mentality pianists?

with regret
A.S.

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:

> > * J. C. Neupert (gegr. 1868), Bamberg
>
> Poor old Neupert. They should give it a rest and retire gracefully.

Hi Paul,

The current holder of
http://www.jc-neupert.de/e/index.htm
"managing director: Wolf Dieter Neupert "
confessed recently in an interview in the journal:

http://www.concerto-verlag.de/
issue #222 October/November 2008 om p.23

"Mein UrgroÃvater Johann Christian
(not to be confused with J.S. Bach's last son J.C.Bach)
Neupert hatte ein merkwÃ¼rdiges Hobby:
Wenn ein neues Klavier verkauft wurde,
musste man meist vom Kunden ein altes Instrument vom
Dach-Boden oder aus der Scheune mitnehmen;
die alten Instrumente wurden dann im Hof gestaplet,
und wenn der Haufen groÃ genug war, einfach angezÃ¼ndet...."
tr:
'My great-grandfather had a odd hobby:
When an new piano was sold,
often he had to recieve back an old instrument
from the garret or out of the barn;
that old instrumets were stapeled in the (factories) yard
and if the bunch big enough,
simply ignited and burnt down...'

among that were some original
http://en.wikipedia.org/wiki/Fortepiano
s of
C.G.SchrÃ¶ter, G.Silbermann, Cristofiori, Stein, Walter & Graf....

I still knew once an old guid-master in piano-building
was proud about that he had combusted even more old instruments
than the old Neupert was able to kill off.

But in the movie
http://en.wikipedia.org/wiki/Amadeus_(film)
"# Tom Hulce casting as Wolfgang Amadeus Mozart"
played on a
http://www.jc-neupert.de/e/instr_2/mozart.htm
as clear visible on the company name-plate of the piano in the film.
Ironically the trade-name
http://www.jc-neupert.de/e/index.htm
was "founded 1868",
but that was 79 years after W.A.Mozart's death.

So far about alleged HIP
http://en.wikipedia.org/wiki/Authentic_performance
claims in that movie on a modern Neupert fortepiano.

Don't you believe it,
alike the concocted 'squiggle' fabrication?

bye
A.S.

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:

> ..but like Hesse's Magic Theatre, it's not for everyone...
...especially not for those that persist in refusing to count beats:

http://www.raeng.org.uk/news/publications/ingenia/issue12/Dain.pdf
http://www.ingenia.org.uk/ingenia/articles.aspx?Index=165
"Tuning
A modern quality piano is not tuned to a mathematically correct
temperament. Each instrument has its own natural optimum temperament
at which it sounds most interesting and pleasant. Temperament may also
need modification to match the hearing of the listener. Older people
tend to hear higher notes at a marginally lower frequency than the
young. Concert tuners will stretch the intervals by as much as an
eighth of a tone per octave in the top registers. The tuner may also
deliberately introduce detuning of the unisons to reduce
(or sometimes even increase)
the decay rate of a particular note exhibiting short sound by putting
the strings slightly out of phase....."

Hence not recommended for the old fortepianos
with desired fast slump regression in loudness.

> How any list not include Christopher Clarke, the Father of Us All,
> as Beethoven said of Haydn?
http://home.planet.nl/~klink507/ENGELS/CLEMENTI.HTM
"In this respect we are also indebted to Christopher Clarke, who
during an `Antwerpiano Symposium', organised by the renowned
fortepiano and harpsichord player Jos van Immerseel, gave detailed
instructions in a paper he presented to the symposium about the
setting up of different types of actions after they had been rebuilt."

Agreed in deed,
he really deserves an place of honour for preserving old pianos.

Last act:
http://www.dieterwunderlich.de/Hesse_steppenwolf.htm
"...magisches Theater nur für Verrückte
Eintritt kostet den Verstand. Nicht für jedermann...."

'...magic theatre - only for madman -
entrance causes lost of mind. Not for everyone....'

drop-curtain:
The show is over.

bye
A.S.

🔗Paul Poletti <paul@...>

🔗Tom Dent <stringph@...>

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> Just a quick closing comment. Your thread started out referring to
> "vibrato" in tuning piano unisons, but most of what you have posted in
> the meantime is about the fact that each individual string of a
> tri-chor note can be at a slightly different pitch when sounding
> alone, yet when sounding with the others they all mode lock and sound
> as one, but with a different decay rate because the mechanism of mode
> locking alters the input impedance on the bridge. In other words, no
> "vibrato".

Exactly, so the definition of a 'mistuned' unison is debatable. I
think most people would only consider it mistuned with audible beats,
which allows for some frequency deviation between individual strings.
Still, this causes a definitely noticeable alteration in timbre and
'envelope', compared to frequencies that are 'dead on'. Which some
people may or may not like.

> The same thing can happen on fortepianos. The three strings can be
> sounding as one together, but if you check just two or one you may
> find they are at a slightly different pitch. These instruments, in my
> experience, always sound better when all three strings are at
> precisely the same pitch. You can either use an external electronic
> reference tone or tune each string apart to the the octave.

It would be interesting to see if there is any historical reference to
this. Historical tuning instructions don't say anything about unisons,
though, do they? They are assumed to be obvious.
It may explain why some old piano recordings sound 'wrong' despite
there being no obvious beats. But what sounds 'better' is also debatable.

Is is possible to describe the audible difference in fortepianos due
to 'locked' strings of different individual frequencies, versus truly
unison strings?
~~~T~~~

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> Exactly, so the definition of a 'mistuned' unison is debatable. I
> think most people would only consider it mistuned with audible beats,
> which allows for some frequency deviation between individual strings.
> Still, this causes a definitely noticeable alteration in timbre and
> 'envelope', compared to frequencies that are 'dead on'. Which some
> people may or may not like.
>
http://www.amarilli.co.uk/piano/kirk.asp
"The strings and bridge constitute a system of coupled oscillators
(Weinreich, JASA, 1975). For a given partial number : It does not
therefore follow that a difference of frequency between the strings
sounded in isolation, results in any 'out of tune' characteristics in
the unison when both strings are sounding, such as beating or
significantly non linear decay. Kirk does not suggest such differences
constitute an 'out of tune' condition in the usual sense......

.....Artist tuners with wide experience will also know that there is
much more to fine unison tuning than the question of decay rate, and
that on many pianos there are likely to be far more pressing reasons
why 'mistuning' between strings would be introduced."

> It would be interesting to see if there is any historical reference
> to this.
http://www.amarilli.co.uk/piano/theory/paradigm.asp
"
"False" beating is not an anomalous "add on" to an otherwise complete
picture of piano tone and tuning. Rather, it is an integral and
inherent part of natural piano tone behaviour, noticed in the
scientific investigation of piano tone at least as far back as 1935
(Wolf and Sette, JASA, 6, 160-168), and mentioned as early as 1907 by
the piano tuner Cree Fischer (Piano tuning a simple and accurate
method). It is at least in part due to the natural behaviour of piano
string motion itself, which is much more complicated than as supposed
by the standard theory for beats. Piano strings do not simply vibrate
polarized in one plane, as the standard theory of beats requires. Nor
can the transverse motions of piano strings in situ be adequately
described by more generalized elliptical polarization. Rather, in
general, the description of transverse piano string motion must allow
for parametric loci. As early as 1943 Schuck and Young (JASA, 15, 1,
1-11) suggested that what we would in the context of piano tuning call
false beat phenomena, could be explained by the rotation of the plane
of vibration of the piano string, and possibly by the transfer of
energy from one mode to another."
>
> Is is possible to describe the audible difference in fortepianos due
> to 'locked' strings of different individual frequencies, versus
> truly unison strings?

....
In addition to the existence of two distinct types of "beating" in
piano tuning, is the fact that bridge coupling between strings allows
for the adjustment of partial decay rates and amplitudes, at least in
unisons, and possibly in other intervals (Weinreich, G, "Coupled piano
strings" JASA, 62, 1474-1484, 1977). The existence of false beats in
the context of bridge coupling, presents an acoustical system in the
actual piano tuning situation, which the standard theory of beats is
simply not able to describe. The description provided by the latter,
even taking into account inharmonicity, is a crude simplification and
approximation for the actual situation. Nonetheless, it has penetrated
the psyche of a good many theorists, would-be theorists, and
practitioners who claim to be good tuners. It is in the nature of the
human mind to attempt to fit the interpretation of experience to
presumed knowledge already established in the mind, and in the case of
the popular paradigm for tuning, learning from the experience of
tuning even correctly perceiving what is actually there - can be
impeded by holding inadequate mental/theoretical models for the
phenomena being encountered. The better theorists generally know from
experience that a natural phenomenon is always likely to be more
complicated than previously supposed, but unfortunately there are many
who never question the popular paradigms.

Provided it is understood that "beating" and "beats" are somewhat
generic terms, it is reasonable to say that (aural) piano tuners tune,
in part, by "listening to beats". However, rather than using the
paradigm of tuners applying beat rates to "aurally estimate" some
theoretically "worked out" tuning, perhaps a more accurate paradigm
would be:

The finest piano tuning is an art, carried out by tuners drawing on a
deep empirical knowledge of piano tone and tuning behaviour. All
current theoretical models for piano tuning are just models. Even if
we improved the model to account for the data we now have for piano
tone behaviour, in the words of Samuel Karlin, The purpose of models
is not to fit the data, but to sharpen the questions."

Qutation from:
(Karlin, S, 11th RA Fischer Memorial Lecture, Royal Society,
20/4/1983, cited in Buchanan, Mark, Ubiquity, Phoenix, 2000)

bye
A.S.

🔗Paul Poletti <paul@...>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

> "False" beating is not an anomalous "add on" to an otherwise complete
> picture of piano tone and tuning. Rather, it is an integral and
> inherent part of natural piano tone behaviour, noticed in the
> scientific investigation of piano tone at least as far back as 1935
> (Wolf and Sette, JASA, 6, 160-168), and mentioned as early as 1907 by
> the piano tuner Cree Fischer (Piano tuning a simple and accurate
> method).

Harrumph! Reminds of "LA Story", where Steve Martin is driving the
British woman around Beverley Hills, and at one point says, with a
touch of admiration and awe in his voice, "Some of these homes are as
old as 1956!"

Try 1824, Johann Lorenz Schiedmeyer, Kurze Anleitung zu einer
richtigen Kenntniss und Behandlung der Forte-Pianos in Beziehung auf
das Spielen, Stimmen und Erhalten derselben (while his "instructions"
may be "short", his title certainly wasn't!). It's essentially a
reprint of Streicher's 1802 book of almost the same name, they were
friends. He adds a lot of technical information Streicher did not
have. He blames false strings primarily on the heavy stringing which
has become popular as of late, and says that when two strings of a
tri-chor sound pure but the third is not, it ruins everything, and the
only thing to do is replace the string. Unfortunately, while he goes
to great lengths to tell you that the piano must be more or less in
tune before properly tuning it, because if you don't, changes made in
the extremes will destroy your temperament octave, he doesn't say
anything about tuning unisons.

I find that an out-of-tune yet mode-locked choir sounds a bit dull
compared to a truly in-tune. The way I usually know that it is
happening is that after tuning the choir to a beatless state, I find
the note to be out-of-tune with its octave from whence I got the
reference to tune it in the first place. Usually upon checking, only
one of the three strings is out with the octave, and when you correct
it individually aginst the octave, upon checking the choir with all
three strings aain, the note sounds much louder and has a better ring.

Ciao,

🔗Carl Lumma <carl@...>

🔗Paul Poletti <paul@...>

🔗Tom Dent <stringph@...>

🔗Carl Lumma <carl@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>

> Some of these homes are as
> old as 1956!"
>
> Try 1824, Johann Lorenz Schiedmeyer, Kurze Anleitung zu einer
> richtigen Kenntniss und Behandlung der Forte-Pianos in Beziehung auf
> das Spielen, Stimmen und Erhalten derselben (while his "instructions"
> may be "short", his title certainly wasn't!). It's essentially a
> reprint of Streicher's 1802 book ....

>.... Unfortunately, while he goes
> to great lengths to tell you that the piano must be more or less in
> tune before properly tuning it, because if you don't, changes made in
> the extremes will destroy your temperament octave, he doesn't say
> anything about tuning unisons.

That was said already in 1511 by
http://en.wikipedia.org/wiki/Sebastian_Virdung
's
http://mdz10.bib-bvb.de/~db/bsb00008398/images/index.html?seite=13
http://www.library.appstate.edu/music/lute/vir1511.html
"The three bass courses each have a thinner octave string, for volume,
and each sancksaytt is doubled at the unison. All of the strings are
sheep gut."
Modern spoken:
There he discusses the inconsistene of partials,
due to improper deviations string-diameters,
in order to enrich the poor harmonic spectra
by further additional aliquot strings for improved resonance.
Appearently J.BlÃ¼thner overtook his aliqout-concept from there.
>
> I find that an out-of-tune yet mode-locked choir sounds a bit dull
> compared to a truly in-tune. The way I usually know that it is
> happening is that after tuning the choir to a beatless state,
How about even with higher partials out of tune?

> I find
> the note to be out-of-tune with its octave from whence I got the
> reference to tune it in the first place.
Exactly observed!

> Usually upon checking, only
> one of the three strings is out with the octave,
Virdung & BlÃ¼thner reccomend to replace such "false" strings
by new ones in order to gain more uniformity.

> and when you correct
> it individually aginst the octave,
That's in their's view barely an compromise,
only tolerable when no new string is avialable without the defect.

> upon checking the choir with all
> three strings aain, the note sounds much louder and has a better
> ring.
Confirmed,
because that slight detuning away from unison improves the sound
of the trichord.

in Virdung's 1511 terms:
Sunder sie mÃ¼ssen eyn Unisonum/
oder eyn gleiche Stym haben/....

Detection of irregularietes in gut-strings:
Schlage dan mit einem Daumen auff die saitte/
also daÃ sie zittern wirt / und prummeÂ´ /
So sie dann zittert / und so vill du der selben saiten/
wiederschlagung oder erscheinung minder sichts.
So vill ist sie dester besser / So viell du ir mer sichts/
So viel ist sye deter bÃ¶ser / die hat dann aynen koz allain/
und heisset die quintsait. Als do stett.

Then there follow some instructions (incl. picture)
in reference to Boethius
how to judge about such strings and theirs
beatings near unison.

bye
A.S.

🔗Andreas Sparschuh <a_sparschuh@...>

🔗Carl Lumma <carl@...>

Andreas wrote:

> > Andreas, it's quite hard to figure out what assertions you
> > are making, if any....
>
> Ok,
> it's easily said in absolute pitches for the tri-chords:
>
> c' 263 (+-~0.25) Hz, the middele_C
> #' 272.5 (+-~0.25)
> d' 294.5 (+-~0.25)
> #' 312 (+-~0.25)
> e' 329 (+-~0.25)
> f' 351 (+-~0.25)
> g' 342.2 (+-~0.25)
> #' 416 (+-~0.25)
> a' 440 (+-~0.25)
> #' 468 (+-~0.25)
> b' 493.5 (+-~0.25)
> c" 526.25 (+-~0.25)
>
> Simply compare two pianos that are both constructed in the
> same way, but each got tuned in different ways:
>
> 1. the one instrument with all 3 strings exacty unison @ center
> value 2. the other with the above deviations +-~0.25 inbetween
> the 3 strings
>
> Quest:
> Which one sounds better in yours ears?

Thanks. I might try that. But I have some questions:

1. Are there any recordings of a piano tuned both ways,
that we can compare?

2. Why do you bother giving the temperament above?
Does the particular temperament matter, or can I apply
the "unison formula" to any temperament?

3. Your unison formula, I take it, is one string of a
triple tuned 1/4 Hz. flat, one tuned dead on, and the
other tuned 1/4 Hz. sharp. Correct?

4. Regarding the unison formula: Does it matter which
strings I tuned sharp or flat, or shall I just follow
the pattern, left = flat, center = dead on, right = sharp?

5. How shall I tune double-strung notes? +0.25 and -0.25?

6. Very important: WHY should this unison pattern be
desirable? Can you explain it in your own words? With
a model behind it, I could better justify spending my
time on this experiment.

thanks,

-Carl

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Andreas wrote:
>
> > > Andreas, it's quite hard to figure out what assertions you
> > > are making, if any....
> >
> > Ok,
> > it's easily said in absolute pitches for the tri-chords:
> >
> > c' 263 (+-~0.25) Hz, the middele_C
> > #' 272.5 (+-~0.25)
> > d' 294.5 (+-~0.25)
> > #' 312 (+-~0.25)
> > e' 329 (+-~0.25)
> > f' 351 (+-~0.25)
> > g' 342.2 (+-~0.25)
> > #' 416 (+-~0.25)
> > a' 440 (+-~0.25)
> > #' 468 (+-~0.25)
> > b' 493.5 (+-~0.25)
> > c" 526.25 (+-~0.25)
> >
> Thanks. I might try that. But I have some questions:
>
> 1. Are there any recordings of a piano tuned both ways,
> that we can compare?
Sorry, not yet.
>
> 2. Why do you bother giving the temperament above?
Because it's easy to set in 5ths
(attend the Helmholtz-notation):

A110 a220 a'440Hz
e'329 ( < 330 := 3*A110)
b"987 := 3*E329
F#185 f#370 f#'740 f#"1480 f#'''2960 ( < 2961 := 3*b"987)
c#"555 := 3*F#185
,,,G#13,,G#26,G#52 G#104 g#208 g#'416 g#"832 g#'''1665(<1665:=3*c#555)
,Eb39 := 3*G#13
Bb117 := 3*,Eb39
f'351 := 3*Bb117
( g/3=65.7 131.4 < )c131.5 c'263 c"526 c'''1052 ( < 1053 := 3*f'351)
( 7*7=49 98 198 < ) g197.1
( 7*7*3=147 294 588 < ) d'589 ( < 591.3 := 3*g197.1 )
a'440Hz ( < 441 := 49*9 = 147*3)

!neoSeptenarius.scl
!
a328/329eb2960/2961F#C#G#1664/1665EbBbf1052/1053c1314/1315g5890/5913d1760/1767a
!
12
!
555/526 ! c#"/c"
589/526 ! d"/c"
312/263 ! eb'/c'
329/263 ! e'/c' ! (5/4)*(1316/1315) ! 3rd ~1.316...Cents sharp
351/263 ! f'/c' ! (4/3)*(1053/1052) ! 4th ~1.645...Cents sharp
370/263 ! f#'/c'
1971/1315 ! (5g')/(5c') ! (3/2)*(1314/1315) ! 5th ~1.317...Cents flat
416/263 ! g#'/c'
440/263 ! a'/c'
468/263 ! bb'/c'
987/526 ! b"/c"
2/1
!

Electronic simulation:
Take an second and third MIDI-piano,
adjust pitch-bend to exactly constant +-0.25Hz beat-rate
and hit at all 3 instruments the same key,
in order to gain the same effect as on a
3-fold stringed real acoustic piano.

> Does the particular temperament matter,
by no means,
you can also use whatever temperament you'd like to perfer.

> or can I apply
> the "unison formula" to any temperament?
In deed.
But in the above one,
the 3rd C-E satisfies for instance:

with
c'- 262.75
c' 263 = e'''-/5 = 1315/5
e'''/5 = 263.2 = 1316/5
c'+ 263.25

for the 5th partial, when taken two octaves down:
c'-*5/4 = 328.4375
e'- = 328.75 = 1315/4 = c'*5/4
e' = 329
c'+*5/4 = 329.0625

that yields an well fusing 3rd C-E.

>
> 3. Your unison formula, I take it, is one string of a
> triple tuned 1/4 Hz. flat, one tuned dead on, and the
> other tuned 1/4 Hz. sharp. Correct?
Yep in deed, that's correct:
Exactly one beat within 4 seconds duration in time.

>
> 4. Regarding the unison formula: Does it matter which
> strings I tuned sharp or flat, or shall I just follow
> the pattern, left = flat, center = dead on, right = sharp?
Good idea, when all 3 strings posess the same inharmonicity
also in the higher partials, then
it's completely up to you
whereever you want to locate that 3 different pitches,
in whatsoever order you prefer that.
But in practice the strings differ in inharmonic behaviour
for the higher partials:

Hence my recommendation:
in order to compensate deviations inbetween the higer partials
among the 3 strings of the same tone:

1.Tune the string with the most lowering inharmonic slight high.
and analogous
2.Tune string with the most arisen inharmonic partials slight low.
vice versa.

>
> 5. How shall I tune double-strung notes? +0.25 and -0.25?
That's up to yours taste.
I prefer to strech the octaves
so that the two outest
stings yield an exact
beatless octave up 2:1 or down 1:2 precisely
but all other combinations turn out wider than an octave,
in a way so that it turns out to be close to the individual
http://en.wikipedia.org/wiki/Image:Railsback2.png
of the actual piano.

>
> 6. Very important: WHY should this unison pattern be
> desirable?
It the synchonous beating inbetween the 3 strings 1
matches the the synchochronous beating of temperament
and vice versa, so that even the frame of the instrumet
vibrates in quaver notes at metronome 60 with
with accent on the 1st beat 4/4 bars.

> Can you explain it in your own words?
The when playing in that interlocking combination of
temperament and within-the-single-note-detuning
the frame of the wohle piano should vibrate
synchronously resonant in quavers,
as pre-imagined by
Schlick, Werckmeister & Jorgenson.
http://www.mmdigest.com/Tech/jorgensen.html
for the metronome speeds of 30, 60, 120 ,240.

bye
A.S.

> With
> a model behind it, I could better justify spending my
> time on this experiment.
>
My model tries to match an Metronome at speeds 30, 60, 120 & 240
by as well by the temperament as well by the detuning of
the 3-chords within a single-tone in order to get rid of
the need of an real metronome.

bye
A.S.

🔗Carl Lumma <carl@...>

Andreas wrote:

> > Does the particular temperament matter,
>
> by no means,
> you can also use whatever temperament you'd like to perfer.

Thanks.

> > 3. Your unison formula, I take it, is one string of a
> > triple tuned 1/4 Hz. flat, one tuned dead on, and the
> > other tuned 1/4 Hz. sharp. Correct?
>
> Yep in deed, that's correct:
> Exactly one beat within 4 seconds duration in time.

Thanks.

> > 4. Regarding the unison formula: Does it matter which
> > strings I tuned sharp or flat, or shall I just follow
> > the pattern, left = flat, center = dead on, right = sharp?
>
> Good idea, when all 3 strings posess the same inharmonicity
> also in the higher partials, then
> it's completely up to you
> whereever you want to locate that 3 different pitches,
> in whatsoever order you prefer that.
> But in practice the strings differ in inharmonic behaviour
> for the higher partials:
>
> Hence my recommendation:
> in order to compensate deviations inbetween the higer partials
> among the 3 strings of the same tone:
>
> 1.Tune the string with the most lowering inharmonic slight high.
> and analogous
> 2.Tune string with the most arisen inharmonic partials slight low.
> vice versa.

OK.

> > 6. Very important: WHY should this unison pattern be
> > desirable?
>
> It the synchonous beating inbetween the 3 strings 1
> matches the the synchochronous beating of temperament
> and vice versa, so that even the frame of the instrumet
> vibrates in quaver notes at metronome 60 with
> with accent on the 1st beat 4/4 bars.

But this justification depends on the temperament's beat
rate. I thought you told me the -0.25 0.0 +0.25 unison
pattern would improve the sound of any temperament. (?)

-Carl

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> > ....synchonous beating inbetween the 3 strings 1
> > matches the the synchochronous beating of temperament
> > and vice versa,...

> But this justification depends on the temperament's beat
> rate.
Fully agreed,
that in my particular case of tuning
the beatings of the temperament and beatings
among the 3 strings got interlocked alike in
an tooth-wheel gear-drive mechanism.

> I thought you told me the -0.25 0.0 +0.25 unison
> pattern would improve the sound of any temperament. (?)
Yep i think so in my own taste,
but also understand Paul's preference of closer
approximation to unison,
as far as the inhomogene partials do allow that in deviation.

Anyhow, independent from the chosen temperature,
the ~+-0.25 technique may help to reduce false beats
inbetween the differences among higher inharmonic partials.

Literature reference
on that unavoidable oscillatory phenomenon:
Blüthner, 1872 Leipzig
http://www.buchhandel.de/detailansicht.aspx?isbn=978-3-923639-94-6
P.233
"...Dagegen kann es sich leicht ereignen,
daß sich eine Saite beim Aufziehen verdreht,
in welchem Falle sie ebenfalls einen unreinen Klang geben wird.
Es bleib in einem solchen Falle kein anderes Mittel übrig,
als eine solche Saite zu entfernen und an ihrer Stelle eine
neue aufzuziehen.
Auch kommen Saiten vor die ein etwas ungleichartiges
Gefüge haben, deshalb keinen reinen Klang ergeben und
nothwendigerweise from Instrument entfernt werden müssen.
Manchmal läßt sich auch, wenn einzelne Saiten zu wenig Gesang haben,
dadurch Abhülfe schaffen, daß man eine schwächere Nummer aufzieht,
die gewöhnlich mehr Gesang giebt; doch kommt es auch vor,
daß man durch eine stärkere Nummer dem Übelstande abhelfen kann..."

'...against that, it can easily happen,
that an string twists in itself contorted.
That ill case results also in an impure dirty sound.
There's no cure for that, nor exists an alternative option:
The sorely-afflicted string has to be removed and
instead replaced by an new one.
Also do occur strings with inhomogenous partials
(disturbed joining), hence gain no pure sound.
Sometimes,
when a single string produces to faint singing,
a change helps to a number weaker (less in diameter of wire),
that results usually in an greater extend in singing;
but also may help to escape from the mischief by
hoisting an number stronger wire...."

Never the less:
But all those well-tried wrinkles
can barely reduce the inhomogene
deviations limited in extent,
without getting completely rid
of an residual disturbance,
that inherently remains,
when compareing real acoustic pianos
versus electronically sterile replica.

Hence i do prefer to
'sugar-coat' that unshirkable impurity
by enriching the sound consistently broder
by ~+-0.25 Hz enhancement in both directions of pitch.

Quest:
Any thoughts about that proposal?

bye
A.S.

🔗Paul Poletti <paul@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
Hi Paul,
> Sound like Piano Celeste to me.
not to be confused with 'Schiemayers'
http://www.celesta-schiedmayer.de/us/index.htm
>
> http://www.organstops.org/c/Celeste.html
"Celestes are tuned to beat at anywhere from 1/2 to about 7 beats per
second."

But i do remain beneath of that at about
~ 1/4=0.25 beats per second,
so that sound of the 3 strings
stay locked within fuse due to the 'pulling-effect':

"If a celeste rank is physically located too close to the on-pitch
rank with which it sounds, the two ranks can pull each other into
tune, ruining the effect of the celeste. The remedy for this is to not
place the two ranks too close together."

http://en.wikipedia.org/wiki/Aliasing
"....aliasing refers to an effect that causes different continuous
signals to become indistinguishable (or aliases of one another)..."

http://en.wikipedia.org/wiki/Anti-aliasing
"...anti-aliasing is the technique of minimizing the distortion
artifacts known as aliasing when representing a high-resolution signal
at a lower resolution..."
...in order to get rid of disturbing to high amplitudes in
higher partials, for pronouncing the lower partials:
That yields an sonorous pilant warm tone,
when compared to the harsher exact trifold-unison,
without the damping of the higher partials.

Alternative:
A compromise inbetween consists in:
Tune the 3 strings so,
that they match exacly beatless inbetween
the first overtone (2:1) the octave
instead of the 3 different zeroth fundamenatal (1:1) pitches.

Simply let bias in inharmonicity among the 3 strings decide
how much detuning among the fundamental they do need,
in order to obtain fitting beatless octaves
out of the residual spectre.

Quest for happiness,
may be more suiting to Paul:
Could you chum up
with tuning barely the octaves beating-less,
while tolerating slight tiny tolerances
inbetween the fundamentals, couldn't you?
At least for dulcet 'romantic' euphonic intonation?

bye
A.S.