Common Cents?

Hi, folks. I have a concern which I'm going to have a go at answering myself:

I've noticed on this list that most posts can be divided into roughly two topics: a) variations on 12-tone scales, and b) non-12-tone scales.

What concerns me slightly is the use of "cents" in both cases to describe scale intervals. It seems to me that a) cents only really fit 12-tone scales, and b) cents are a linear "difference" of what is actually a geometric relationship - they don't scale correctly for higher octave divisions.

I've been doing basic scale calculations with a spreadsheet, and cents don't seem to be a useful concept in that context. I found that a simple error percentage calculation was a better guide to how consonant intervals sound (or don't). I wrote a long article on this topic, with several tables, but thought better of inflicting it on the list - this is the version that cuts to the chase!

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.

So, that's my point - cents is of use in a historical context and with primitive* instruments, but it doesn't help when searching for the lost chord..! 8{

Cheers,

Brian Thomson, London UK
bnt@email.com

* You know what I mean, right?
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Unfortunately, scholarly usage (beginning with A. Ellis)
has stuck us with only one type of cent -- that of
which there are 1200 in a 2:1 octave. 19-tET has steps of
63.16 cents, and 22-tET has steps of 54.55 cents. The 3:2
("fifth") is 7.22 cents flat in 19-tET, and 7.14 cents sharp
in 22-tET.