Re : Common Cents?

Brian Thomson wrote:

>
> The table below compares the perfect fifth (3/2) to the "fret positions"
> calculated under 12ET (7), 19ET (11) and 22ET (13), using two different
> error measurements: cents difference, or % error. For 19ET and 22ET, I've
> chosen to divide the octave up into 1900 and 2200 respectively, which makes
> the numbers easier to compare by eye, but not affecting the result.
>
> "dec.ratio" is the ET ratio calculated using 2^(n/12), while "Rcents" is
> 3/2 coverted to cents using log2(3/2) and multiplied by 1200, 1900 and 2200
> as appropriate. %diff is the error ratio between dec.ratio and 3/2 (1.5) .
>
> Scale ETcents Rcents c.err dec.rat %diff 330Hz beat
> 12ET 700 701.96 1.96 1.4983 -0.113% 0.372Hz
> 19ET 1100 1111.43 11.43 1.4938 -0.416% 1.373Hz
> 22ET 1300 1286.92 -13.08 1.5062 0.413% 1.363Hz
>
> By the cents measurement, 22ET looks worse than 19ET, while using the %
> error measurement, it's marginally better. If we take beat frequency
> between the ideal and the actual as a measure of fit, the beat frequencies
> follow the %diff ratios (19ET has approx 4 times the beat frequency as
> 12ET). I've tried generating these intervals using CoolEdit, and as best as
> I can measure on screen and hear the relationship holds true in practice.

Of course, if you choose separate ways to measure a cent in 22ET and 19ET
it seems impossible to compare 13.08 22ET-cents with 11.43 19ET-cents, just
by looking at it.

Wim Hoogewerf