Relationship of terms, was Important Question, was lucky break

[Mike and Jane Spalding]

Alan,

Others have begged us not to respond with real math, but, unfortunately, mathematics is the only language that can properly convey the answer to your question.

The ratio of the frequencies of two notes which are one cent apart is the 1200th root of two, or 1.000578.  In other words, one cent is one 100th of an equal tempered half step expressed as a logarithm.  In other words, if you start on A4, and increase the frequency by one cent, one hundred times, you will end up on equal tempered A#4.  If you increase the frequency by one cent, twelve hundred times, you will end up on A5.  (Assuming no inharmonicity.)  In your example, 111.1Hz is about 17.4 cents sharp of 110Hz.

Beats, on the other hand, are caused by the interference of waves of nearly equal frequency, and the beat rate will be equal to the difference in Hz.  In your example, the two strings will beat at a rate of 1.1 Hz  (111.1-110)

Clear as mud?  I think maybe Reblitz has a good explanation.

hope that helps,

Mike Spalding RPT

----- Original Message ----- 
From: Alan R. Barnard <mathstar@salemnet.com>
To: Pianotech <pianotech@ptg.org>
Sent: Wednesday, September 18, 2002 10:37 PM
Subject: Relationship of terms, was Important Question, was lucky break


> Aha, aha, aHA! That's where I'm going with this ... how do Hz, Cents, and
> Beat Rates all interplay?
> 
> Does a certain percentage (cents) of distance between two tones determine
> the beat rate without regard to the Hz of those tones, or does it change
> with scale location, i.e., frequency?
> 
> Unisons: Small changes in the treble area make a very fast beat because the
> frequency changes drastically, compared to strings in the bass BUT what we
> call "cents" is obviously proportional and/or exponential to frequency
> because any given difference in cents means much a higher difference in
> frequency (Hz) in the treble than in the bass.
> 
> Anyone have an oscilloscope?
> 
> Experiment: One string of a bichord base note at, say, 110 Hz Fundamental (a
> theoretically perfect A3) and its companion string at 111.1 Hz. This is a
> one percent difference. Is that one "cent?" What is the beat/roll rate?
> 
> Now compare that to two strings in the treble, one at 2093.004
> (theoretically perfect C7) and another in that unison at exactly 2113.934
> Hz--also a one percent difference.
> 
> Will those two misaligned unisons each beat at the same rate exactly? Does
> each pair differ by the same "cents" if they differ by exactly the same
> percentage of Hz?
> 
> Is my brain just spinning cobwebs, or does anyone else wish they had a
> clearer understanding of the relationships of these terms?
> 
> Alan R. Barnard
> Unraveling in Salem, MO
> 
> ----- Original Message -----
> From: "sid blum" <sid@sover.net>
> To: "Pianotech" <pianotech@ptg.org>
> Sent: Wednesday, September 18, 2002 8:59 PM
> Subject: Re: Important Question, was lucky break
> 
> 
> >
> > ----- Original Message -----
> > From: Kevin E. Ramsey
> >
> >   "  Well, Alan, unless I'm wrong (In which case I'm sure I'll hear about
> > it), in the area of the upper temperament, the difference between A440 and
> > A441 is approxiamately 4 cents, so 442 would be eight cents sharp.
> >     That being said, one cycle at A0 would be about 64 cents at the
> > fundemental level, according to Jim Coleman Sr. So I suppose that one
> cycle
> > per second at the top of the piano would probably be a fraction of a cent.
> > Just this month he mentioned that, and remarked that it gave a whole new
> > light on lower bass tuning, and quibbling over a few tenths of a cent. "
> > ********************************************************
> >
> > This reminds me of a thought I had recently while reading various posts
> > relating to ETD s.  Would it not be useful to have an ETD that could also
> > read out in the language of frequency (cycles per second)?
> >
> > I'm way over my head here, but I'm thinking a device such as this could
> > relate to beat rates.
> >
> > sb
> >
> > _______________________________________________
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> 
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