Yahoo Tuning Groups Ultimate Backup mills-tuning-list Harmonics at other than integer frequency ratios.

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I don't believe uniformity of the string is the important factor: it is the stiffness of real wires attached to sound boards by being mechanically clamped at their ends (stretched over a fret, pressed down by a finger, whatever) which is responsible for the anharmonicity of string instruments. I suppose if one could weld tiny bearings of negligible moment of intertia to the ends of a wire, and then pull on the bearings to generate the string tension, one might be able to create the kind of boundary condition at the ends of the wire which would eliminate this effect. However, if one did just this, most listeners would probably complain: "That doesn't sound like a real ....."

When one synthesizes musical tones by additive Fourrier synthesis this is exactly the kind of sound you get: perfectly harmonic, but somehow not quite believable. Most listeners are used to, and tend to prefer, the sounds of real, anharmonic instruments which have these theoretically objectionable "defects", otherwise there wouldn't be so many synths on the market based on waveform sampling of "real" instruments rather than using Fourrier synthesis from harmonic partials - or so it seems to me.
Clif Ashcraft

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From: James Kukula[SMTP:kukula@synopsys.com]
Sent: Friday, January 03, 1997 1:09 AM
To: aaaabb1@peabody.sct.ucarb.com
Subject: integral overtones

Natural systems have all sorts of wierd overtone sets. Nice uniform strings
will generate integer overtones. Nonlinear systems - systems with distortion,
like overdriven loudspeakers - also seem to generate integer overtones.

Systems with just one degree of freedom seem to be constrained to move
periodically, which forces integer overtones, as has already been pointed
out. One degree of freedom means a two dimensional phase space (the space of
combinations of position and velocity). In three or more dimensions one can
embed arbitrary graphs, but in two dimensions the possibilities are cut back
severely. In a similar way it seems like the trajectory of a simple
oscillator just doesn't have the topological freedom to fold around in
strange ways. I might well be wrong about this. (I should hunt around in the
literature but most of that stuff goes way over my head!)

But the tendency of nonlinearity to create integral overtones seems to go
beyond one dimensional systems. In a relatively calm ocean, waves of all
different frequencies run over each other freely. But once waves start
cresting, then distinct shapes start to appear and move through space
together. This sort of phase coherence is again a periodic motion that
implies integer overtones.

It does seem that nature has a certain prejudice in favor of integers!

Actually our common use of integers to count distinct objects seems related
to the kind of phase coherence created by nonlinearity. Nonlinearity makes
things clump together and split apart, and that's why there are things to
count. (And to do the counting!) Huh, and that's really the root of the
distinction between digital and analog circuitry - digital circuits use
distortion to make nice distinct pulses. (It's getting too late at night!)

Jim

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🔗linusliu@HK.Super.NET (Linus Liu)

>Perhaps a string player on this list would like to comment: has anyone
ever encountered a ''nice uniform string''?

I cannot think of any possibility of myself or any string player who
would not get upset if s/he finds himself out of tune. To any string
player, a note can only be either in tune or out. Also to him/her, any
note played can only have one single intonation. This is the only way
I can comprehend. This is quite different from the hearing on a
bell, for instance, several "notes", meaning several notes each with
an unique intonation, which I/anybody can distinctly hear at the same
time on a single played "note" (_a_ bell).

So, the fact that the higher harmonics at other than integer ratios than
the fundamental hardly bothers us.

However, the piano is different. I like to ask everyone on the list that
did not have such experience: when a very low note on the piano is sounded,
it might be hard to tell which note it is, probably due to that "enharmonic"
effect. But what I, and probably anyone else, do is to either listening down
from a higher, recognizable, note up the scale, so to "know" this note, and
then to remember (the intonation of) it, or "look" at the position of the
note one the keyboard, so one actually "know" what note it is.

But after this exercise, did any one of you still find the "wrong" intonation
of this note a problem in any way?

What I find is that the ear will then "avoid" hearing this note. But the
subsequent interpretation and appreciation of the sound of this note, is
more a mental process than anything else.

But one thing is for sure. I only THINK about only one single intonation
of that note, or any note for that matter.

One thing is very sure. The intonation I hear on any note played by a
string player is that same intonation everyone else hear. My audience
always come and tell me which note I played in tune or out, exactly as
I know they are. This happens because I play so often for my friends'
weddings. Even more obviously so singing in a choir (vibrato avoided
for heavens sake).

Linus.

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🔗PAULE <ACADIAN/ACADIAN/PAULE%Acadian@...>

Danie wrote:
>In his all natural harmonics string quartet, Chronos Kristalla (1990,
>Material Press), La Monte Young found it necessary to tune the harmonics
>themselves, not the fundamentals in order to get the level of consonance he
>requires. In fact, when the open strings of his quartet are then played
>with one another, beating is abundant although the intervals between their
>natural harmonics are essentially beatless. (All of this is said ignoring
>the role of bow technique).

Much as in the trombone example, you are confusing the issue of harmonic
tones and harmonic partials. The act of touching the string to make it
produce a harmonic will increase its tension and therefore not result in the
same pitch as an overtone of the open string. A steadily bowed string
exhibits a periodic waveform; therefore, the partials are harmonic.

>>Fourier proved that any periodic waveform can be expressed as the sum
>>of sine waves whose frequencies form an EXACT harmonic series.

>I quote from the Encyclopedic Dictionary of Mathematics (MIT 2nd Ed 1993)
>''Fourier also stated (without rigorous proof) that an arbitrary function
>could be represented by trigonometric series, a statement that gave rise
>to subsequent developments in analysis. (Vol 1, p 620)''

He did indeed also state that fact, much disbelieved in his time.

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Harmonics at other than integer frequency ratios.

🔗clucy@cix.compulink.co.uk (Charles Lucy)

🔗"Ashcraft AC (Clif)" <aaaabb1@...>

🔗linusliu@HK.Super.NET (Linus Liu)

🔗PAULE <ACADIAN/ACADIAN/PAULE%Acadian@...>

🔗Gary Morrison <71670.2576@...>