Perfect inharmonicity

Some rather startling (at least to me, initially) conclusions I found
yesterday evening...

Go back to the formula for the overtones

f_n = n f_0 sqrt(1 + B n^2)

They said 'f_0 is the fundamental', but that can't be exactly the
case: f_0 is what the fundamental would be if there were no
inharmonicity. The real fundamental is
f_1 = f_0 sqrt(1 + B).

Now consider two notes with 'ideal fundamentals' f_a and f_b played on
strings of inharmonicity B_a and B_b. We suppose they have
nearly-coincident m'th and n'th partials f_a,m and f_b,n. The true
frequencies will be

f_a,m = m f_a sqrt(1+B_a m^2)
f_b,n = n f_b sqrt(1+B_b n^2)

Now we have m f_a approximately equal to n f_b. The coincidence
condition will include the ratio of square roots, which is how the
inharmonicity mucks things up. Even if you get partials 'm' and 'n' to
coincide, if you go up to 2m and 2n you get a different ratio of
square roots - you can't get the overtones an octave above to work out.
But there is one exception! That is: if B is a constant times f_0^2.
Say B = f_0^2/f_B^2, where f_B is some very large frequency.

Suppose B_a = f_a^2/f_B^2, B_b = f_b^2/f_B^2. Now for the two
coincident partials we get the square root factors

sqrt(1 + m^2 f_a^2/f_B^2)
sqrt(1 + n^2 f_b^2/f_B^2)

but these are the same because m f_a = n f_b!

So, providing that the inharmonicity B increases as the square of the
frequency, there is *no* conflict in tuning two notes such that
partials n/m, 2n/2m, 3n/3m, etc. etc. all match. What you get is just
the naive relations m f_a = n f_b between the 'ideal fundamentals'.
Although the frequencies are stretched they mesh together.

Now look at the B formula again. We had
B = metal properties * d^4/(l^2 T)

But T = stuff * d^2 f_0^2 l^2, so we get

B = stuff * d^2/(f_0^2 l^4)

Now remember 'Pythagorean scaling' - the length of a string is inverse
to its frequency: if this is the case, f_0^4 l^4 can be taken nearly
constant.

B = stuff * d^2 f_0^2 /(f_0^4 l^4)

So, if the string diameter and the scaling proportion (f_0 l) stay
constant up the scale, you get exactly the right relation B = stuff *
f_0^2 !
Basically: The most naively constructed instrument with length
doubling every octave and the same string thickness everywhere, has
just the right distribution of inharmonicity to allow perfectly
coinciding partials.

In reality, string diameter gently decreases and (f_0 l) gently
increases up the scale, for good musical/technical reasons, so
inharmonicity doesn't increase so fast as would be required for a
perfect match.

Although, if you look at the graph at
http://i30.photobucket.com/albums/c348/mireut/sandersonB.png
the slope of a lot of the lines is nearly the 'ideal' factor of 4 per
octave in pitch (eg increase from 0.01 to 10 over 5 octaves).

Any note or register that 'sticks out' above those smoothly rising
lines will cause trouble because its higher partials will be
increasingly sharp (with increasing n) compared to the partials at
similar frequency of other notes it should make harmony with.

That's why although bass notes have quite low inharmonicity measured
by B they cause problems, since they don't maintain the precipitous
decline that would be consistent with the rest of the scale. A more
useful measure would be B/f_0^2.

The slow decrease of B/f_0^2 through the tenor and middle also
explains (hopefully) why Bill Bremmer's '2:1', '4:2' and '6:3' octaves
are different. If the higher string is thinner and/or has
proportionately longer scaling, its partials won't 'keep up' with the
lower note's so it would have to be increasingly sharpened to get a
match of higher and higher partials.

Do stop me if you know all this already...
~~~T~~~

Sorry for the delayed response. I certainly didn't know
that. I knew that all scale designers strive for smoothly-
increasing inharmonicity curves, but I had always puzzled
as to why.

-Carl

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> Some rather startling (at least to me, initially) conclusions
> I found yesterday evening...
>
> Go back to the formula for the overtones
>
> f_n = n f_0 sqrt(1 + B n^2)
>
> They said 'f_0 is the fundamental', but that can't be exactly
> the case: f_0 is what the fundamental would be if there were no
> inharmonicity. The real fundamental is
> f_1 = f_0 sqrt(1 + B).
>
> Now consider two notes with 'ideal fundamentals' f_a and f_b
> played on strings of inharmonicity B_a and B_b. We suppose they
> have nearly-coincident m'th and n'th partials f_a,m and f_b,n.
> The true frequencies will be
>
> f_a,m = m f_a sqrt(1+B_a m^2)
> f_b,n = n f_b sqrt(1+B_b n^2)
>
> Now we have m f_a approximately equal to n f_b. The coincidence
> condition will include the ratio of square roots, which is how the
> inharmonicity mucks things up. Even if you get partials 'm' and
> 'n' to coincide, if you go up to 2m and 2n you get a different
> ratio of square roots - you can't get the overtones an octave
> above to work out. But there is one exception! That is: if B is
> a constant times f_0^2. Say B = f_0^2/f_B^2, where f_B is some
> very large frequency.
>
> Suppose B_a = f_a^2/f_B^2, B_b = f_b^2/f_B^2. Now for the two
> coincident partials we get the square root factors
>
> sqrt(1 + m^2 f_a^2/f_B^2)
> sqrt(1 + n^2 f_b^2/f_B^2)
>
> but these are the same because m f_a = n f_b!
>
> So, providing that the inharmonicity B increases as the square of
> the frequency, there is *no* conflict in tuning two notes such
> that partials n/m, 2n/2m, 3n/3m, etc. etc. all match. What you
> get is just the naive relations m f_a = n f_b between the 'ideal
> fundamentals'. Although the frequencies are stretched they mesh
> together.
>
> Now look at the B formula again. We had
> B = metal properties * d^4/(l^2 T)
>
> But T = stuff * d^2 f_0^2 l^2, so we get
>
> B = stuff * d^2/(f_0^2 l^4)
>
> Now remember 'Pythagorean scaling' - the length of a string is
> inverse to its frequency: if this is the case, f_0^4 l^4 can be
> taken nearly constant.
>
> B = stuff * d^2 f_0^2 /(f_0^4 l^4)
>
> So, if the string diameter and the scaling proportion (f_0 l) stay
> constant up the scale, you get exactly the right relation
> B = stuff * f_0^2 !
> Basically: The most naively constructed instrument with length
> doubling every octave and the same string thickness everywhere, has
> just the right distribution of inharmonicity to allow perfectly
> coinciding partials.
>
> In reality, string diameter gently decreases and (f_0 l) gently
> increases up the scale, for good musical/technical reasons, so
> inharmonicity doesn't increase so fast as would be required for a
> perfect match.
>
> Although, if you look at the graph at
> http://i30.photobucket.com/albums/c348/mireut/sandersonB.png
> the slope of a lot of the lines is nearly the 'ideal' factor of
> 4 per octave in pitch (eg increase from 0.01 to 10 over 5 octaves).
>
> Any note or register that 'sticks out' above those smoothly rising
> lines will cause trouble because its higher partials will be
> increasingly sharp (with increasing n) compared to the partials at
> similar frequency of other notes it should make harmony with.
>
> That's why although bass notes have quite low inharmonicity measured
> by B they cause problems, since they don't maintain the precipitous
> decline that would be consistent with the rest of the scale. A more
> useful measure would be B/f_0^2.
>
> The slow decrease of B/f_0^2 through the tenor and middle also
> explains (hopefully) why Bill Bremmer's '2:1', '4:2' and '6:3'
> octaves are different. If the higher string is thinner and/or has
> proportionately longer scaling, its partials won't 'keep up' with
> the lower note's so it would have to be increasingly sharpened to
> get a match of higher and higher partials.
>
> Do stop me if you know all this already...
> ~~~T~~~