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From: Ron N. on 3 Aug 2010 13:34
On Aug 2, 11:17 am, fatalist <simfid...(a)gmail.com> wrote:
> The fundamental frequencies of open strings on a tuned guitar form
> some well defined ratios, like 4/3
>
> The combined signal produced by all strings is thus supposed to be a
> periodic signal if the guitar is well tuned
>
> Can be done in time-domain, I believe, without any FFT but with some
> trickery...

If the guitar is tuned to an equal-temperament scale, then the
ratio between pitches is irrational (being the 12th root of 2),
not a well defined integer ratio.

Furthermore, due to physical effects such as string stiffness
and non-zero diameter, the ratios between overtones are often
not exact integer ratios either, but slightly "inharmonic".
That's one of the reasons why physical pianos are "stretch"
tuned to slightly different pitches than any set of simple
mathematic ratios.

With a strobe tuner or synchronous Lissajous display, you can
actually see the non-exact-harmonic-ratio overtones in lower
notes.

This is one of the reasons why I think that FFT-based pitch
estimation methods may be better than either autocorrelation
or state-space methods for a few classes of musical instrument
sounds.


IMHO. YMMV.
--
rhn A.T nicholoson d.0.t C-o-M
http://www.nicholson.com/rhn/dsp.html
From: fatalist on 3 Aug 2010 17:06
On Aug 3, 1:34 pm, "Ron N." <rhnlo...(a)yahoo.com> wrote:
> On Aug 2, 11:17 am, fatalist <simfid...(a)gmail.com> wrote:
>
> > The fundamental frequencies of open strings on a tuned guitar form
> > some well defined ratios, like 4/3
>
> > The combined signal produced by all strings is thus supposed to be a
> > periodic signal if the guitar is well tuned
>
> > Can be done in time-domain, I believe, without any FFT but with some
> > trickery...
>
> If the guitar is tuned to an equal-temperament scale, then the
> ratio between pitches is irrational (being the 12th root of 2),
> not a well defined integer ratio.
>
> Furthermore, due to physical effects such as string stiffness
> and non-zero diameter, the ratios between overtones are often
> not exact integer ratios either, but slightly "inharmonic".
> That's one of the reasons why physical pianos are "stretch"
> tuned to slightly different pitches than any set of simple
> mathematic ratios.
>
> With a strobe tuner or synchronous Lissajous display, you can
> actually see the non-exact-harmonic-ratio overtones in lower
> notes.
>
> This is one of the reasons why I think that FFT-based pitch
> estimation methods may be better than either autocorrelation
> or state-space methods for a few classes of musical instrument
> sounds.
>
> IMHO. YMMV.
> --
> rhn A.T nicholoson d.0.t C-o-M
>  http://www.nicholson.com/rhn/dsp.html

Forgive me if I'm wrong, but as far as I know most of the adjacent
guitar strings are tuned to be in perfect fourth, that is a pitch
ratio of 4/3, VERY CLOSELY approximating the equal temperament scale
based on dividing an octave into 12 equal intervals on a logarithmic
frequency scale.
In fact, as far as I remember, this is the basis of the entire modern
music system.

Don't know about all guitar strings sounding together, but with just 2
strings perfect fourth is readily identifiable both perceptualy and
using analysis methods like phase-space reconstruction.
From: Richard Dobson on 3 Aug 2010 17:52
On 03/08/2010 22:06, fatalist wrote:
..
>
> Forgive me if I'm wrong, but as far as I know most of the adjacent
> guitar strings are tuned to be in perfect fourth, that is a pitch
> ratio of 4/3, VERY CLOSELY approximating the equal temperament scale
> based on dividing an octave into 12 equal intervals on a logarithmic
> frequency scale.
> In fact, as far as I remember, this is the basis of the entire modern
> music system.
>
> Don't know about all guitar strings sounding together, but with just 2
> strings perfect fourth is readily identifiable both perceptualy and
> using analysis methods like phase-space reconstruction.

I am not a guitarist, so I am not quite prepared to commit myself 100%
to this, but: the difference between a pure fourth and the ET fourth is
very slight - under 2 Cents. When not using an eletronic tuner,
guitarists generally refine/verify the tuning by sounding a unison
(sometime the octave) between an open string and an adjacent fretted
string. They also use natural harmonics in a similar way. In other
words, whatever they tune the open string to has to be such that any
fretted octaves are true. The focus for this is especially for the G/B
strings, for the reasons given before.

Re stiffness of strings: the tension is vastly less on a guitar than on
a modern piano (where it can be measured in tons rather than pounds) -
not much scope for string-bending and vibrato on a piano. So any
stretching of harmonics will be very slight indeed - probably less even
than on a harpsichord. May be more of an issue on a bass guitar. The
very act of fretting a note on a guitar stretches the string very
slightly, so it would not be unreasonable to say that a guitar is tuned
within a somewhat hand-wavy margin of a cent or two, here and there! And
of course much hinges on the guitar maker getting those frets ~exactly~
in the right place and exactly at the same height; something that sadly
cannot be guaranteed in all guitars at all price points, all the time.

Richard Dobson
From: Jerry Avins on 4 Aug 2010 00:30
On 8/3/2010 5:52 PM, Richard Dobson wrote:
> On 03/08/2010 22:06, fatalist wrote:
> .
>>
>> Forgive me if I'm wrong, but as far as I know most of the adjacent
>> guitar strings are tuned to be in perfect fourth, that is a pitch
>> ratio of 4/3, VERY CLOSELY approximating the equal temperament scale
>> based on dividing an octave into 12 equal intervals on a logarithmic
>> frequency scale.
>> In fact, as far as I remember, this is the basis of the entire modern
>> music system.
>>
>> Don't know about all guitar strings sounding together, but with just 2
>> strings perfect fourth is readily identifiable both perceptualy and
>> using analysis methods like phase-space reconstruction.
>
> I am not a guitarist, so I am not quite prepared to commit myself 100%
> to this, but: the difference between a pure fourth and the ET fourth is
> very slight - under 2 Cents. When not using an eletronic tuner,
> guitarists generally refine/verify the tuning by sounding a unison
> (sometime the octave) between an open string and an adjacent fretted
> string. They also use natural harmonics in a similar way. In other
> words, whatever they tune the open string to has to be such that any
> fretted octaves are true. The focus for this is especially for the G/B
> strings, for the reasons given before.
>
> Re stiffness of strings: the tension is vastly less on a guitar than on
> a modern piano (where it can be measured in tons rather than pounds) -
> not much scope for string-bending and vibrato on a piano. So any
> stretching of harmonics will be very slight indeed - probably less even
> than on a harpsichord. May be more of an issue on a bass guitar. The
> very act of fretting a note on a guitar stretches the string very
> slightly, so it would not be unreasonable to say that a guitar is tuned
> within a somewhat hand-wavy margin of a cent or two, here and there! And
> of course much hinges on the guitar maker getting those frets ~exactly~
> in the right place and exactly at the same height; something that sadly
> cannot be guaranteed in all guitars at all price points, all the time.

Don't ignore bridge (and nut) compensation which, because thy are always
straight, are unlikely to be exact.
http://www.lutherie.net/saddle_angle.html

Jerry
--
Engineering is the art of making what you want from things you can get.
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From: Richard Dobson on 4 Aug 2010 05:19
On 04/08/2010 05:30, Jerry Avins wrote:
....
>> within a somewhat hand-wavy margin of a cent or two, here and there! And
>> of course much hinges on the guitar maker getting those frets ~exactly~
>> in the right place and exactly at the same height; something that sadly
>> cannot be guaranteed in all guitars at all price points, all the time.
>
> Don't ignore bridge (and nut) compensation which, because thy are always
> straight, are unlikely to be exact.
> http://www.lutherie.net/saddle_angle.html
>
> Jerry


Indeed. But not quite always - at least on electric guitars it is common
to have a separate screw-adjustable saddle for each string. I have never
seen one on an acoustic guitar though.


Richard Dobson
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