millioctaves

Great idea! I've been using these for years, since before I
discovered the rest of the world used cents. Silly things,
surprised they caught on. It would certainly make things easier
for me if everyone else came round to my way of thinking, although
this does run into problems with reality.

As to the notation, I suggest "moct" so that you can switch to
"oct" when the intervals get big enough. 2.735 oct makes a lot
more sense to me than 3282 cents. I prefer mET to MET: the latter
would imply megaoctaves! Microoctaves also have their uses: the
MIDI Tuning Standard works to 5 mu-oct, doesn't it?

Apart from removing the 12TET bias, octaves are easier to
calculate : p log2(f/f_0); f0*2^p. Step sizes of equal
temperaments can be worked out really easily -- sometimes in your
head.

I've also been using octaves to denote absolute pitches, with
f_0z. Once you know that middle C corresponds to about 8.04
oct, you know that 10.128 oct must be somewhere above the treble
clef. What does 1119 Hz mean? Setting f_0Hz may be better,
equivalent to subtracting 4 oct, and around the threshold of
hearing. The fractional part can also be replaced with a note
name: 8.04 oct becomes 8C, 8.78 oct becomes 8A. This can be
generalised for any tuning system you like.

Well, that's the way I see it, anyway.

Graham

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Charles Lucy posed a whole big slew of questions about viewing a
vibrating string from various angles and in various lights. Unfortunately
if he was trying to come to some sort of point with those questions, I'm
afraid I totally missed it, despite having read them twice.

Best I can tell his last questions were something of a summary:

> Could the integer frequency ratios be approximations?
> Can you "prove" that the integer frequencies ratios is the
> only possible mathematical model to describe what you
> observe?

The idea that strings, and most air columns as well, used in common
musical practice have precisely harmonic partials is indeed only an
approximation. It is also very good approximation.

You can see that readily by decomposing and then recreating such a tone
with absolutely perfectly harmonic partials, which I personally have done.
Without a doubt it's certainly noticeably different, but its effect upon
harmony is far less than, for example, chorus effect. The relative
amplitudes of the partials are almost certainly more significant to the
perceived character of harmonies than such minuscule pitch deviations.

That however is not even close to true for characteristically
nonharmonic instruments like xylophones, bells, timpani, and is only
marginally true for the piano.

One of these days I'll have my internet/web act together and post some
strictly harmonic tones with their "original" versions.

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Charles Lucy wrote:

> I maintain that I have found a better mathematical model
> to map "ghosttones"

So far you have only followed this up with questions. I'm
waiting to see your better model. I hold that real strings
have partials which are nearly but not exactly harmonic.
It's up to you to show that this is not so, and that the
proper model is built on the value pi, and that this
justifies your tuning system for plucked and hammered
string instruments, and that this justifies your tuning
system for other instruments.

Matt Nathan

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