tuning lower notes - partials explained

[Matt]

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At 11:47 PM 10/12/04 -0400, you wrote:

>               You know, I've been reading all these wonderful, 
> informative posts on tuning and octaves here, gleaning what I understand, 
> but I really don't know what 6:3 or 8:4 octaves means.
>
>
>Julia,
>Reading, Pa

Hi Julia:

Those ratios refer to the "coincident partials" of both notes in an interval.

It helps to know what the partial series for a vibrating string is:

1st: fundamental (or the actual notes itself)
2nd: 1 octave up
3rd: 1 octave + 1 5th up
4th: 2 octaves up
5th: 2 octaves and a major 3rd up
6th: 2 octaves and a 5th up
7th: non-musical for all intents and purposes but roughly 2 octaves and a 
minor 7th up
8th: 3 octaves up

...the series goes on but the first 8 are the most important and relevant.

Now: EVERY vibrating string has these components in it (and in fact some 
hammer voicing techniques can be used to accentuate certain 
partials).  Another fact: There is a certain drop off point, as we go down 
the keyboard in pitch, where we don't hear the fundamental partial anymore 
and our mental aural picture of bass string pitch is really comprised of 
its partials only.

So: if every vibrating string has this series of components, then it 
follows that if you play any interval (let's say an octave C2 - C3), there 
will be points where the partials for each string overlap, or are 
coincident, and are consequently louder because of being doubled.  Here's 
the series for both pitches:

        1     2    3    4     5     6    8
C2:  C2  C3  G3  C4  E4  G4  C5
C3:  C3  C4  G4  C5  E5  G5  C6

As you can see, both strings have many partials in common and when you play 
the octave on the piano you can actually hear those pitches.  For example, 
a popular partial to listen to for this octave is the 6:3 or the 6th 
partial of C6 and the 3rd partial of C3, which is G4.  If you actually play 
G4 briefly and then play the C2-C3 octave it will help you to hear the G4 
component which should be prominent.  When tuning the octave at the 6:3, 
the idea is to have the G4 beating really slow or not at all.  The trick 
is, on some pianos, if you tune the 6:3 to be clean, then the 8:4 will beat 
unacceptably.  Then you have to make a judgement call...

You will find these coincident partials on any interval you use to tune - 
3rds, 4ths, 5ths, 6ths, etc.  And it helps to learn how to listen to them 
while tuning.

Hope this is helpful.

-Matt

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