common cents

[PHIL SLOFFER, MA067 855-1244]

Hi All

I think my own confusion about cents has to do with whether the number is
referring to an exact pitch or to the size of an interval.  I think you
have to keep both in mind to understand what is going on.

If you start from middle C4 and go up by half steps in equal temperament
it looks like this.

NOTE      C4   C#4  D4   D#4  E4   F4   F#4  G4   G#4  A4
CENTS     0    100  200  300  400  500  600  700  800  900       etc.
SAT       0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0

So from C4 to G4 is 700 cents, an equal tempered fifth.
And from D4 to A4 is also 700 cents.  900-200=700

A pure or just fifth is 702 cents.

>From C4 to E4 is 400 cents, an equal tempered 3rd.
>From E4 to G#4 is also 400 cents. 800-400=400

A pure or just third is 386 cents.

Now then the SAT says all note are 0.0 if they do not vary from the
mathematically correct equal temperament.  So if you tuned C4 to 0.0 and then
wanted to tune a pure fifth above you would set the SAT to tune G4 +2.0 cents,
that is to say 702 cents.

Or if you wanted to tune a pure third then C4 0.0 and E4 -14.0
400 cents - 386 cents = 14

Now I know there ain't no Santa.  So of course the SAT will most likely have
different numbers on display because of the partial it is listening to.


Even more confusing is comparing temperaments from two different sources.
Suppose you found a table for 1/4 comma meantone that says something like this.

NOTE  C        D         E         G         A
CENTS 0        193       386       696.5     889.5
SAT   0.0      -7.0      -14.0     -3.5      -10.5

>From another source it say this:

NOTE  C        D         E         G         A
CENTS 10.5     203.5     397       707       900
SAT   +10.5    +3.5      -3.0      +7.0      0.0

They are both right of course.  The intervals are the same size.  It just
depends on where you start.  With A or C.  That is to say do you want A=440
or C=261.6255.   And of course once you put it into the machine you can reset
the machine up or down as needed to get the pitch you want.

As to converting cents to beats I think others have answered that well enough.
I think it has also been mentioned before that people offen confuse beats and
cents when talking about pitch.

What bugs me is when people talk about the pitch being A440 (or A415.3) and
use cents that are for C261.6255  And they will talk about temperaments being
different when they are in fact the same.  Most of Owen's work uses cents that
go with C=261.6255.  But if you tune aurally and want a A=440 you have to redo
the temperament scheme before you can get the results you want.

It is no wonder people get confused when they tune and things don't work out
right.

Sorry if this is too long for some of you.

Phil Sloffer
psloffer@ucs.indiana.edu