Cents making sense

[Niklas Eliasson]

Richard Moody asks:

> ... OR is there a way to figure beat rates from cents?

Oh, definitely! Without going into any deep details, let us suppose
we have, as Richard M describes, a table where we can see the cent
values for a certain temperament, for example:

A  --- 0
A# --- 98
B  --- 196
...
and so on.
For different c:s, c=0,100,200,300 and so on, we have equal temperament.
This is the way you often find in the literature, when a specific
temperament is described.

>From above, we can say that B is four cents flat, related to equal temp,
since in equal temp, B is 200.

The question now is: How do we compute actual (theoretic!) beat rates??

If we don't want to use log-expressions etc, we can equivalentely use
this formula for calculating the beats of a given fifth:

     beat=f*(3-2*2^(c/1200))

Here, f is the frequency of the lower note, and c is the ammount of
cents of the higher one.
If, for example we have c=700, i.e equal temp, then we get

     beat=f*(3-2*2^(700/1200)) = f*0.003386

which for, say f=220 for A, gives

     beat=220*0.003386 = 0.745 beats per second

If we have a table that says "E = 695 cents", we use

     beat=220*(3-2*2^(695/1200))=2.7 beats per second

----------

Similarly, for fourths:

     beat=f*(3*2^(c/1200)-4)

and for major thirds

     beat=f*(4*2^(c/1200)-5)

where, as above, f allways is the freq of the lower note!


So, Richard, if you read that a third is flat by 3.5 cents compared
to equal temp, then take the third formula above and put

 c=400-3.5 = 396.5   (since c=400 means equal temp)

Thus,

     beat = f*(4*2^(396.5/1200)-5) = f*0.0295

So, for your middle C, with f=261.6 Hz, the beat rate turns out to

     beat = 262.6*0.0295 = 7.7 beats per sec.

For A=440 you get beat=0.0295*440=13 beats per sec, and so on.

This way, we can easily put up tables for tuning (in theory)

We must remember, though, that it is just theory where the inharmonicity
is not taken in. Of course, we can make formulas for that too, but
that is really senseless...however, important is that we are able to
compute an UPPER BOUND for how much the inharmonicity affects the
temperament! But I will not go into that discussion here! (I have tried
it before....) That is also a discussion about instrument building and
how to design the scales etc.

Now, for the more advanced use of my formulas, consider the table

A  --- 0
A# --- 98
B  --- 192
C  --- 312
C# --- ..
D  --- ..
D# --- ..
E  --- 701

Question: What is the beat rate between C and E here, if A=220???

Use the formula

     beat=f*(4*2^(c2/1200)-5*2^(c1/1200))

where now c1=312 and c2=701

Put this in and we get

     beat = 2.04..... beats per sec.

------

After all, this is theory! It is completely useless to use more than
say three digits in the results we get. If we get a beat rate
A that is 13.76254, it is just frustrating to deal with. Its enough
to say 13.8.

And about equal temperament. I know there is never so in practice,
due to inharmonicity, or we can put it:

All temperaments are equal, but some are more equal than others...

(Couldnt resist paraphrasing Orwell.)



Greetings to all

Niklas beenatthemathdepartementheretoolong Eliasson

Linkoping, Sweden