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From: Bo Parker 
Subject: Re: Tuning and Tuners

brian@devil.sbil.co.uk (Brian Agnew) wrote

> In article EDB@fc.hp.com, kalstein@cnd.hp.com (Mike Kalstein) writes: 
> 
> >1. You're tuning to just intonation when you really want to tune >to
> equal temperament, so you have to do minor adjusting afterwards >by
> playing chords in various keys.. > 
> 
> I've heard of this 'temperament' tuning before, but I don't know what
> it is. Can someone please explain ?

Hi

This was discussed some time ago. This was my attempt to explain 
equal-temperament:


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Greetings.

Equal-tempered tuning was invented, as has been stated, to make it
possible to play a quantized instrument (like guitar, piano, etc.) in
any key, ``reasonably'' in tune, without having to have some huge
number of keys/frets/whatever per octave. The main compromise in equal
temperament is in the third and sixth intervals. For example, I bet a
lot of you guitarists out there have noticed the way that when you get
your guitar in tune (by ear) to sound great on an E major chord, it
will then sound a little ``out'' on a G major chord. To illustrate 
how this sort of discrepancy arises, consider the major third ``in
harmonics'' on a guitar string that occurs around (actually a little
behind) the fourth fret. At this harmonic, the string vibrates in 5
equal parts. This note is a ``just'' major third up from the double
octave harmonic at the 5th fret (at which the string vibrates in 4
equal parts). Thus the ratio of frequencies for a ``just'' major third
is 5/4 = 80/64. This interval sounds good to most people, and is
actually what most people hear as a ``correct'' major third. Let's use
harmonics to derive a major third another way now. Tune a string to a
note, say A. Now tune another string exactly one octave down from the
7th fret harmonic of this string. This will then be an E. Repeat this
process. This produces a B. Repeat again. This produces an F#. Repeat
once more. This produces a C#, which is ``another kind of'' major
third (with intervening octaves) up from our original A. What is the
frequency ratio? It's (3/2)^4 = 81/16, which reduces to 81/64, after
eliminating two octaves. (We can eliminate octaves because they're the
only interval that stays perfectly in tune with its harmonics.)
Anyway, this interval sounds pretty crummy to most people (it's too
wide). Let's compare the various major third intervals now.

``just''  5/4 = 1.2500
method 2  81/64 = ~1.2656
equal tempered = 2^(4/12) = ~1.2599

You will notice that the equal tempered major third is between the just
major third and the ``method 2'' major third, but it's still pretty far
off of the ``correct'' just interval.

Let's carry this procedure on to produce the next A. We have to do it
a total of 12 times to get back to A. This gives (3/2)^12.  If this
procedure were to produce in-tune octave(s), then this number would
have to turn out to be a power of 2. It's actually equal to about
129.75. So it's close to a power of 2 (128) but not quite. These are
the sort of discrepancies that make equal temperament necessary.

There are people for whom the compromise in the major third that is
caused by use of equal temperament is actually uncomfortable.
20th-century composer Harry Partch was an example of such a person. He
composed music using (I think) 42 notes per octave. I am on the
borderline with respect to this: usually my ear can accept the
equal-tempered major third, but sometimes (especially on a crummy
guitar, even if it's in tune), it just sounds totally off.

Now for the matter of ``tuning to harmonics.'' By this I mean matching
the 7th fret harmonic on the 5th string to the 5th fret harmonic on
the 6th string, and so forth for the string pairs that are 5 frets
apart, and matching the 4th fret harmonic on the 3rd string to the 5th
fret harmonic on the 2nd string. If tuning to harmonics this way
worked, then the number ((4/3)^4)*(5/4) would have to be equal to 4
(which is the frequency ratio between the low E on the 6th string and
the E on the open 1st string - a double octave). Actually we get

((4/3)^4)*(5/4) = ~3.951

which is pretty far off.  This is because each intervening interval is
a little off (and is off in the same direction), and so the
out-of-tuneness adds up with each successive string pair. What
equal-temperament does, in effect, is to distribute this
out-of-tuneness among all string pairs, though (necessarily) not
equally.

Note that this discussion does not even take into account the
psychoacoustic stuff about tuning the low notes flat and tuning the
high notes sharp to make them sound ``in tune.'' The relationships
among frequency, pitch, temperament, timbre, volume, in-tuneness,
overtones, ``combination tones,'' and so forth, are very complex, and
I personally have only scratched the surface of it. Why, for example,
does an augmented triad (e.g. C E G#), which is made up of only
``consonant'' intervals (major thirds, a minor sixth), sound harsh as
compared to a simple major triad (C E G)? Anybody?

That's my (long-winded) $0.02.

-Bo Parker
parker_b%aplvax.span@fedex.msfc.nasa.gov

I do not speak for my employer, whoever that is.

``...the razor inside, sir...jerk the handle...''

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> Secondly, both my guitar teacher and myself have had problems with
> Korg tuners (mine flicks all over the place). My teacher now has a
> Piccato (sp?), which works really well, but I can't find these
> anywhere. Can anyone tell me about them, or recommend a make ?

I use a Wittner GT-2, and I am very pleased with it.

See ya

-Bo Parker
bo_parker@fbpmac.msfc.nasa.gov (preferred)
parker_b%aplvax.span@fedex.msfc.nasa.gov

I do not speak for my employer, whoever that is.

``...the razor inside, sir...jerk the handle...''