magnetic field

[John Woodgate]

I read in sci.electronics.design that Jon Harris
<jon_harrisTIGER@hotmail.com> wrote (in
<116qa476ll1bqce@corp.supernews.com&gt about 'DIGITAL GUITAR AUTO-TUNER
PROJECT', on Mon, 25 Apr 2005:

However, if I fret some of the strings to play a different chord where,
for example, a string that was the third now becomes the fifth, it is
horribly out of tune! So while for strumming a single chord, just
intonation is great, I have to use a compromise closer to
equal-temperament if I want to actually play a song.

Well, yes, that's precisely why equal temperament was invented.
--
Regards, John Woodgate, OOO - Own Opinions Only.
There are two sides to every question, except
'What is a Moebius strip?'
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk

 

[Jon Harris]

"Jerry Avins" <jya@ieee.org> wrote in message
news:fL-dndOIUI0zkfDfRVn-3A@rcn.net...

In a way analogous to a bell's "clang tone", the harmonic structure
changes rapidly when the string is first plucked. Don't measure the
first part of the note, and don't try for such a long measurement that
the note fades away.

My needle-based digital tuner reacts quite quickly. I can see the note appear
sharp as it is first plucked, then watch it settle in to a fairly constant
reading for several seconds until it fades away. In this case, it is up to me
as the user to ignore that initial transient reading and wait for the steady
state. Doing so is quite natural.

 

[Bob Monsen]

dhaevhid wrote:

Here are some resources:

CircuitCellar... there is a guitar tuner article that uses a
microcontroller. The article describes the entire thing, and gives you a
schematic and a listing of the assembler code.

LCD, it's a snap, given the reams of free code available to drive those
little 16 character one line displays. HD44780 LCDs are cheap. You can
drive them with 11 datalines in 8 bit mode, and 7 in 4 bit mode, and 6
if you just hardwire the write pin high.

Here is a reference: http://ouwehand.net/~peter/lcd/lcd.shtml

They are kind of painful to program, so use a microcontroller with an
on-chip debugger.

I've had some trouble using that CircuitCellar design, however. He
assumes that the first couple of waves are the fundamental, but this
doesn't seem to be true, at least for my guitar. I think a DSP design,
using some simple digital filters, would make more sense. DSPs are
getting pretty cheap. You could tune all the strings simultaneously...


this is the exact reason why im still "hunting" the elusive precise
algorithm on how to do this project the correct way. almost every
project done( atleast those i see on the web) do not consider the
harmonics that would surely cause trouble... i saw some samples though
that used an FFT, IIR and all those stuff but i am yet to learn how to
do all of those in assembly language. i did a project using those
functions in matlab but its an included function in the library. so
its quite a degree more difficult this time to do all the FFT, FIR,
and IIR "hand written" on assembly.

A digital filter is quite simple. It is just a bunch of multiplies and
divides on each sample. However if you have to do the multiply in
software, you'll never make it fast enough, at least with the tiny
microcontrollers that are out there. A PIC 16 series requires between
500 and 700 instructions for a multiply. Using a Zilog Z8 Encore, or a
PIC 18 clocked at 20MHz, you could probably do 6 simultaneous IIR
filters using the same A/D channel (there are web pages that will write
the 'C' code for you, given the frequencies and passband width.
Translating that into asm is trivial.)

By filtering a 10Hz passband for each string, it's possible you will be
able to do the zero crossing thing, and then both select the string, and
give guidance on which way to tune as relative error. However, that is a
guess, since I haven't tried it.

---
Regards,
Bob Monsen

 

[John Woodgate]

I read in sci.electronics.design that Al Clark <dsp@danvillesignal.com>
wrote (in <Xns96437FC08729aclarkdanvillesignal@66.133.129.71&gt about
'DIGITAL GUITAR AUTO-TUNER PROJECT', on Mon, 25 Apr 2005:

the G string on my guitar was actually vibrating at two different
frequencies that straddled the desired center.

Loose wrapping.
--
Regards, John Woodgate, OOO - Own Opinions Only.
There are two sides to every question, except
'What is a Moebius strip?'
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk

 

[Bob Monsen]

Jerry Avins wrote:

Ville Voipio wrote:

...

Nononononnononoooooo! If you tune it to the "even-tempered" scale,
the result is very bad-tempered. All the fifths are bad, all the
thirds are bad.


Ah, come now! Even in the best of guitars, the frets are located by "the
rule of nineteen". Pianos and fretted instruments have tuning
similarities (although you can't "bend" a piano).

Apparently, the very high and very low notes on pianos are tuned with a
larger than 2:1 octave. The ratio can be up to 2.025:1. The claim(1) is
that that this is caused by beat matching, trying to match the
fundamental to the 2nd harmonic, which is slightly off due to the
stiffness of the strings.

Thus, it may not make sense to do precise electronic tuning on stringed
instruments. I always find that my guitar sounds better when I match the
harmonic on the 5th fret of the bottom string to the top string, and
then interpolate. That makes the high E somewhat sharp (but not nearly
as sharp as a piano, due to the lower string tension.)

(1) Musical Acoustics, Donald Hall, pg 188.

---
Regards,
Bob Monsen

 

[John Woodgate]

I read in sci.electronics.design that Guillaume <"grsNOSPAM at
NOTTHATmail dot com"@?.?.invalid> wrote (in
<426d2b10$0$21146$7a628cd7@news.club-internet.fr&gt about 'DIGITAL GUITAR
AUTO-TUNER PROJECT', on Mon, 25 Apr 2005:

As to how the above input stage, based on a filter and a saturation
stage, translates in the frequency domain (in other words, how
the spectrum of the original signal is transformed), I'll let you
think about it.

Thanks for the explanation. It gels with my thinking on the subject.

It resembles, but is not quite like simply looking
at zero-crossings - because the saturation on the signal actually
tends to "ignore" the harmonics, whereas simple zero-crossing
analysis has to deal with them.

Indeed: a very cogent point.

All in all, this is a working approach and it's much simpler than any
sophisticated DSP analysis you might try.

You said it! Digits are NOT always the answer. Particularly FFT.
--
Regards, John Woodgate, OOO - Own Opinions Only.
There are two sides to every question, except
'What is a Moebius strip?'
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk

 

[Paul Keinanen]

On Mon, 25 Apr 2005 16:59:21 GMT, Ben Bradley
<ben_nospam_bradley@frontiernet.net> wrote:

In
sci.electronics.design,sci.electronics.misc,comp.dsp,comp.arch.embedded,
On Sun, 24 Apr 2005 17:29:00 GMT, no-one@dont-mail-me.com (Robert
Scott) wrote:

On Sun, 24 Apr 2005 14:30:37 +0300, Paul Keinanen <keinanen@sci.fi
wrote:

....to add a frequency specific constant
value to a phase accumulator, which will address a sine look-up table
to get the sine wave samples that are then multiplied (mixed) with the
microphone signal to get some kind of frequency/phase detector..

The trouble with phase detectors in the presence of noise is that they
only work if you are close. The indication for "way off" is useless.
It is like using a strobe. The pattern is clear only if the pitch is
within a certain range.

This is where multiple methods come in handy. A low-pass filter and
zero-cross detection gets you in the ballpark, then your method (looks
like a PLL using a phase-accummulator DCO) gets good accuracy.

Yes indeed.

Use a 32 or 64 point FFT (with about 10 Hz bins) to detect the lowest
frequency. This can be done in a few hundred milliseconds.

After this decision, you are within about 10 Hz of the actual
frequency. If you have enough memory, buffer the first tap on the
string and rescan the samples to determine the frequency error.

If no data memory is available, analyze the next tap on the string to
determine the frequency error.

Paul

 

[Paul Keinanen]

On Mon, 25 Apr 2005 07:39:02 +0000 (UTC), rhn@mauve.rahul.net (Ronald
H. Nicholson Jr.) wrote:

This is not so obvious as you will notice there
are nasty transients.

Not sure what you mean here. You do need to make sure your FFT output
looks like it might represent a musical note of interest, and not just
background noise between notes.

The Fourier analysis assumes that you have a constant amplitude
_repeatable_ waveform, unfortunately a string instrument does not
generate such signals, so you have to analyze some part after the
initial transient.

Paul

 

[Jon Harris]

"John Woodgate" <jmw@jmwa.demon.contraspam.yuk> wrote in message
news:RXSgN4lpxSbCFwp0@jmwa.demon.co.uk...

I read in sci.electronics.design that Jon Harris
jon_harrisTIGER@hotmail.com> wrote (in
116qa476ll1bqce@corp.supernews.com&gt about 'DIGITAL GUITAR AUTO-TUNER
PROJECT', on Mon, 25 Apr 2005:

However, if I fret some of the strings to play a different chord where,
for example, a string that was the third now becomes the fifth, it is
horribly out of tune! So while for strumming a single chord, just
intonation is great, I have to use a compromise closer to
equal-temperament if I want to actually play a song.

Well, yes, that's precisely why equal temperament was invented.

I know. I was just giving an example that was guitar-specific in response to a
point that someone else made that equal temperament was bad.

 

[Jerry Avins]

John Woodgate wrote:

I read in sci.electronics.design that Jerry Avins <jya@ieee.org> wrote
(in <H96dnVy_aYL0nfDfRVn-gw@rcn.net&gt about 'DIGITAL GUITAR AUTO-TUNER
PROJECT', on Mon, 25 Apr 2005:

He was quite taken with the selectable temperaments on my recently
acquired Yamaha "piano".


Can this feature be retrofitted to s.e.d contributors?

Even though I have no idea what s.e.d. stands for, I doubt you can
implement selectable intonation without getting into the firmware. In a
YDP-233, one can chose among equal temperament (the default), or pure
major, pure minor, Pythagorean, mean tone, Werkmeister, and Kirnberger,
all with any base note. You can also transpose any number of half tones
and pull the pitch of A440 in about .2 Hz increments from 427 to 453. It
doesn't sound like the Steinway it replaced (my daughter has that now),
but it's fun to play with as well as on. It will record two tracks and
has a MIDI interface. More than I need.

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ

 

[Jerry Avins]

Bob Monsen wrote:

Jerry Avins wrote:

Ville Voipio wrote:

...

Nononononnononoooooo! If you tune it to the "even-tempered" scale,
the result is very bad-tempered. All the fifths are bad, all the
thirds are bad.



Ah, come now! Even in the best of guitars, the frets are located by
"the rule of nineteen". Pianos and fretted instruments have tuning
similarities (although you can't "bend" a piano).


Apparently, the very high and very low notes on pianos are tuned with a
larger than 2:1 octave. The ratio can be up to 2.025:1. The claim(1) is
that that this is caused by beat matching, trying to match the
fundamental to the 2nd harmonic, which is slightly off due to the
stiffness of the strings.

Thus, it may not make sense to do precise electronic tuning on stringed
instruments. I always find that my guitar sounds better when I match the
harmonic on the 5th fret of the bottom string to the top string, and
then interpolate. That makes the high E somewhat sharp (but not nearly
as sharp as a piano, due to the lower string tension.)

(1) Musical Acoustics, Donald Hall, pg 188.

A piano's "stretch" over its 7.5 octaves is the better part of a
semitone. By attaching discrete weights to the strings, most of the
inharmonicity can be removed. (The windings on the low strings end well
short of the bridges, reducing inharmonicity of the second partial.) A
piano so doctored is tuned without stretch. It sounds awful!

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ

 

[Jerry Avins]

Robert Scott wrote:

On Mon, 25 Apr 2005 11:18:06 -0400, Jerry Avins <jya@ieee.org> wrote:

Re: inharmonicity in strings:


I know that you are right, but it puzzles me. It seems to me that the
common bridge should enforce harmonicity, just as it locks together the
slightly detuned piano doubles and triplets. (Exact tuning makes the
note loud ans it's decay rapid. Slight detuning softens the attack,
hoarding energy for better sustain. The same is true of a 12-string guitar.)

It is a real challenge to build 5.0000 and 5.0001 MHz oscillators on the
same chassis, even with crystals, that will actually beat. The
evaporation thickness/rate monitor I wrote of in another thread worked
around that problem.


Yes, there is in fact a "lock-in" phenomenon that happens through the
interaction at the bridge. But typical amounts of inharmonicity, at
least in piano strings, exceeds the lock-in range. Inharmonicity is
easily measured by precisely measuring the pitch of individual
overtones and comparing them. Another way to verify inharmonicity is
to observe the signal on a scope. If the overtones were all locked to
the fundamental, the overall shape of the signal would be constant,
just decaying in amplitude. Perhaps the individual overtones would
decay at different rates. But it would be quite clear that the
overtones are frequency locked. But that is not in fact what you will
observe. If the scope is triggered by the fundamental, then the
overtones will appear as higher-frequency components that are "riding
to the left" as compared to the fundamental.

I agree. One point of quibble: even if the overtones were frequency
locked, the wave shape would still change with time. Different overtones
have different decay rates.

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ

 

[John Woodgate]

I read in sci.electronics.design that Jerry Avins <jya@ieee.org> wrote
(in <brydnYP4-My62_DfRVn-3w@rcn.net&gt about 'DIGITAL GUITAR AUTO-TUNER
PROJECT', on Mon, 25 Apr 2005:

Even though I have no idea what s.e.d. stands for,

It stands for a very great deal, and it's the newsgroup you are
participating in.
--
Regards, John Woodgate, OOO - Own Opinions Only.
There are two sides to every question, except
'What is a Moebius strip?'
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk

 

[Ville Voipio]

In article <426d2b10$0$21146$7a628cd7@news.club-internet.fr>, Guillaume wrote:

By comparing the frequency with a few preset ranges, the tuner can
even guess what the string it is you're trying to tune, and
automatically give you how far away you are from the nominal
frequency for this string.

Thank you for the thorough explanation!

The tuners I've been using seem to be able to guess the tone within
one half, i.e. 5.9% in frequency. This seems to work pretty well,
even though the detection is sometimes inconviniently slo.

However, I've always suspected the tuner measures the frequency
directly and after that finds the nearest note from a LUT and
then calculates the remainder. There is a LED bar display
to show which note (A, A#, B, C, ...) is playing and then an
analog meter to show the deviation from the even temperament
(+-50 cents).

The tuner is quite fine with violins, flutes, and even piccolo flutes,
but not very good with cellos, double basses, bass viols or other
low instruments. It does find the correct note, but the detection
is then very sensitive to higher-frequency noise.

As to how the above input stage, based on a filter and a saturation
stage, translates in the frequency domain (in other words, how
the spectrum of the original signal is transformed), I'll let you
think about it. It resembles, but is not quite like simply looking
at zero-crossings - because the saturation on the signal actually
tends to "ignore" the harmonics, whereas simple zero-crossing
analysis has to deal with them.

But wouldn't a combination of an AGC and a hysteretic comparator
be still better? The saturating stage does give some false
zero crossings even though it filters out a lot. But shouldn't
hysteresis filter all false crossings out, if the sum of
amplitudes of harmonics is smaller than the hysteresis?

---

PLL, anyone? DPLL? In a way a PLL would be the right way to do this,
as the frequency under measurement changes slowly. Locking is,
as usual, a big challenge.

---

An interesting problem, indeed!

- Ville

--
Ville Voipio, Dr.Tech., M.Sc. (EE)

 

[Jerry Avins]

Guillaume wrote:

John Woodgate wrote:

Meanwhile, no commercial tuner that I know of uses an FFT.


How do they work, then? (The answer 'Very well' is not acceptable.)


Most of them use a simple design. You can find one on CircuitCellar,
and that's pretty much how this is done in commercial products.

It consists of an input stage, which is basically a good low-pass filter
filtering everything above the maximum fundamental frequency it's
supposed to deal with (probably something like 1000 or 1500 Hz), usually
a 2nd order active filter. Then it's followed by a comparator set with
some hysteresis, which can also be an amplifier based on some AOP with
a lot of gain - so that the AOP clips the signal, which is easily
transformed into a digital signal with a schmitt trigger, for instance.
This circuit basically extracts the fundamental frequency of the input
signal with a reasonable usability.

Then the comparator's output can be dealt with in various ways.
Some can be rather crude (just measuring the frequency of the resulting
digital signal), some are more clever, and I like the one that's used
in the CircuitCellar project. The comparator's output goes to a digital
I/O pin of a microcontroller, of course set as an input. The algorithm
used consists of measuring the delay between two consecutive raising
edges - but this is not all. To make sure the measure is meaningful,
several consecutive measures are compared, and only if we get a few
(like 10, for instance) consecutive measures that are close enough
to one another, do we consider this is the fundamental frequency.
The latter is computed from the period, using for instance an average of
the 10 given "meaningful" past measures.

By comparing the frequency with a few preset ranges, the tuner can
even guess what the string it is you're trying to tune, and
automatically give you how far away you are from the nominal
frequency for this string.

As to how the above input stage, based on a filter and a saturation
stage, translates in the frequency domain (in other words, how
the spectrum of the original signal is transformed), I'll let you
think about it. It resembles, but is not quite like simply looking
at zero-crossings - because the saturation on the signal actually
tends to "ignore" the harmonics, whereas simple zero-crossing
analysis has to deal with them.

I've agreed so far, but not here. The zero crossings of the comparator's
input and output had better coincide in time (with offset allowed for
hysteresis). Those squiggles away from the crossings will be suppressed,
but they don't matter anyway.

All in all, this is a working approach and it's much simpler than any
sophisticated DSP analysis you might try.

Some wouldn't even call it DSP, but I processed many signals that way
for many years.

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ

 

[Jon Harris]

"Jerry Avins" <jya@ieee.org> wrote in message
news:brydnYP4-My62_DfRVn-3w@rcn.net...

John Woodgate wrote:
I read in sci.electronics.design that Jerry Avins <jya@ieee.org> wrote
(in <H96dnVy_aYL0nfDfRVn-gw@rcn.net&gt about 'DIGITAL GUITAR AUTO-TUNER
PROJECT', on Mon, 25 Apr 2005:

He was quite taken with the selectable temperaments on my recently
acquired Yamaha "piano".

Can this feature be retrofitted to s.e.d contributors?

Even though I have no idea what s.e.d. stands for, I doubt you can
implement selectable intonation without getting into the firmware.

s.e.d. = sci.electronics.design. I think this was a joke, playing off the word
"temperament". Get it?

 

[Ville Voipio]

In article <H96dnVy_aYL0nfDfRVn-gw@rcn.net>, Jerry Avins wrote:

Ah, come now! Even in the best of guitars, the frets are located by "the
rule of nineteen". Pianos and fretted instruments have tuning
similarities (although you can't "bend" a piano).

Depends on the fretted instrument. The idea of fixed frets is
a new one, older instruments tend to have movable frets.
My viol (viola da gamba) has tied frets, and I really have
to move them depending on the temperament. Actually, the
deviation from even temperament is very visible (some
frets are closer to each other than others). And talking about
bending... some of the frets are actually non-horizontal
to give better temperament.

Pianos are sometimes tuned in even temperament. Sometimes not,
many tuners like to make them slightly tempered some way or
another. It is a common misconception that the even temperament
was invented by J.S.Bach (Wohl-temperierte Klavier). The even
temperament was well known in the baroque era -- well known to be
a bad compromise. What Bach discovered was yet another temperament
system which allowed him to play in all keys. The keys sound
fine but different.

I want to make it clear that I don't have any golden ears,
which would be able to detect the frequency down to microhertz.
But the fact is that I have to tune my frets to suit the
needs and temperament of the rest of the band. In that task
a tuner makes life much easier (without the golden ears I
can only say something is wrong

I know it sounds a bit odd, but early music (especially baroque)
players use auto-tuners a lot. For example, tuning a harpsichord
without a tuner takes much longer. It is still possible with some
experience on listening to the weak harmonic beats, but slower.
Or, keeping a viol consort in tune during a longer concert is
much easier with some fixed frequncy reference.

aside> Do you know Roland Hutchinson? (He was quite taken with the
selectable temperaments on my recently acquired Yamaha "piano".)

Yes, I know him (not personally, though). No, I am nowhere near
that level.

(I just wonder if there was anything not off-topic in this posting?)

- Ville

--
Ville Voipio, Dr.Tech., M.Sc. (EE)

 

[John Woodgate]

I read in sci.electronics.design that Jerry Avins <jya@ieee.org> wrote
(in <Toidnai6YvCE1vDfRVn-iA@rcn.net&gt about 'DIGITAL GUITAR AUTO-TUNER
PROJECT', on Mon, 25 Apr 2005:

Winding adds far more mass than stiffness, making the string more
harmonic. Some materials are better than others. Fine silver is used
for violins and cellos. Gold is be better.

Why would gold be better? If it's density, depleted uranium should be
even better!
--
Regards, John Woodgate, OOO - Own Opinions Only.
There are two sides to every question, except
'What is a Moebius strip?'
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk

 

[Spehro Pefhany]

On Mon, 25 Apr 2005 21:27:45 +0100, the renowned John Woodgate
<jmw@jmwa.demon.contraspam.yuk> wrote:

I read in sci.electronics.design that Jerry Avins <jya@ieee.org> wrote
(in <Toidnai6YvCE1vDfRVn-iA@rcn.net&gt about 'DIGITAL GUITAR AUTO-TUNER
PROJECT', on Mon, 25 Apr 2005:

Winding adds far more mass than stiffness, making the string more
harmonic. Some materials are better than others. Fine silver is used
for violins and cellos. Gold is be better.

Why would gold be better? If it's density, depleted uranium should be
even better!

The numbers I saw last time I looked at were only an RCH apart at
~19.3 gm/cm^3 (along with tungsten). Osmium, rhenium, iridium and
platinum are a fair bit higher; plutonium a bit higher.

http://www.science.co.il/PTelements.asp


Best regards,
Spehro Pefhany
--
"it's the network..." "The Journey is the reward"
speff@interlog.com Info for manufacturers: http://www.trexon.com
Embedded software/hardware/analog Info for designers: http://www.speff.com

 

[Bob Monsen]

Jerry Avins wrote:

Bob Monsen wrote:

Jerry Avins wrote:

Ville Voipio wrote:

...

Nononononnononoooooo! If you tune it to the "even-tempered" scale,
the result is very bad-tempered. All the fifths are bad, all the
thirds are bad.




Ah, come now! Even in the best of guitars, the frets are located by
"the rule of nineteen". Pianos and fretted instruments have tuning
similarities (although you can't "bend" a piano).


Apparently, the very high and very low notes on pianos are tuned with
a larger than 2:1 octave. The ratio can be up to 2.025:1. The claim(1)
is that that this is caused by beat matching, trying to match the
fundamental to the 2nd harmonic, which is slightly off due to the
stiffness of the strings.

Thus, it may not make sense to do precise electronic tuning on
stringed instruments. I always find that my guitar sounds better when
I match the harmonic on the 5th fret of the bottom string to the top
string, and then interpolate. That makes the high E somewhat sharp
(but not nearly as sharp as a piano, due to the lower string tension.)

(1) Musical Acoustics, Donald Hall, pg 188.


A piano's "stretch" over its 7.5 octaves is the better part of a
semitone. By attaching discrete weights to the strings, most of the
inharmonicity can be removed. (The windings on the low strings end well
short of the bridges, reducing inharmonicity of the second partial.) A
piano so doctored is tuned without stretch. It sounds awful!

Jerry

That is interesting. The book I referenced mentioned another possibility
for why the stretch occurs. The human ear has far less ability to
discriminate pitch at low and high frequencies than it does in the
middle range. The other possibility was that the extra bit was required
to overcome this inability, forcing the ear to hear an octave. However,
the text also states that studies by Backus indicate that the mechanism
of string inharmonicity accounts for most of the stretching..

I guess you have listened to these pianos. Do the high strings sound flat?

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Regards,
Bob Monsen