Rating ET's by JI approximation

I've been working on a page to help people decide on a tuning system for
their music and instruments. It contains an analysis of equal temperaments
up to 12tet (eventually to 36). The point is to see which ones have the
best approximations of JI intervals without having too many notes to be
practical. Have a look:

http://www.organicdesign.org/peterson/tuning/equal.html

My question is: Have I chosen a useful reference scale? I used all ratios
of numerator and denominator <=13 and their inversions. I know there is a
128 cent gap each side of the unison. Is that a good thing or a bad thing?
If you think I should fill it, what with? If I allow 15/14ths then I leave
myself open to multiplying every numerator or denominator by 3 and getting a
bunch of notes I really don't care about. Currently, I am only allowing
multiplication by 2 because it gives the inversions of all my ratios. What
is the meaning of the fabricated notes less than 128 cents that harmonize
with scale degrees, but not with the original unison? How should they
relate to scalar analysis?

Complete, it is not, but feel free to alert me to spelling, math, or HTML
errors, or information that is just plain wrong. Please do not link to this
page until it is finished.

Thanks!

---
Glen Peterson
Peterson Stringed Instruments
30 Elm Street North Andover, MA 01845
(978) 975-1527
http://www.organicdesign.org/peterson

Glen,

One of my main topics of discussion (and interest) in my ~4 years on this
list and for ~6 years before that has been precisely this. One issue I have
with your results is the fact that you're associating some very complex
ratios with intervals that are too far away to be heard as such. For
example, the major sixth in 12-tET you call an 11-cent out-of-tune 22:13.
But in most consonant contexts it will be heard as a 16-cent out-of-tune
5:3. A general rule of thumb (Harry Partch's Observation One) is that the
"field of attraction" (in cents) of a ratio is inversely proportional to the
size of the numbers involved in the ratio. In this case, that would mean
that if 16 cents mistuning renders a 5:3 unrecognizable, then 3.7 cents
mistuning will render a 22:13 unrecognizable. I have studied the mathematics
of ratio-series and find that Observation One comes out as a good
approximation due to the greater density of competing ratios near, say,
22:13, compared with those near 5:3. My work on harmonic entropy has allowed
me to make these judgments more precise and determine, under a given
assumption on the ear's tuning resolution (I usually use 1% based on various
research), the probability of associating a particular ratio with a given
interval.

So you might think you should just assign each interval its most likely
ratio-interpretation by this measure, but there are pitfalls to doing that
too. One of them is the "consistency" issue that I keep bringing up. Since
you will most likely want to approximate not only JI dyads but chords as
well, you must be sure that your approximations are consistent with one
another. Although there aren't any examples of that with the ratios you've
suggested, try using similar methods to give JI approximations for ET triads
and you'll start running into problems. The solution is to find the odd
limit that each ET is consistent through (see my paper, Patrick Ozzard Low's
paper, or Paul Hahn's web pages) and ignore ratios using odd numbers higher
than the limit.

>I know there is a
>128 cent gap each side of the unison. Is that a good thing or a bad thing?

I think it's most definitely a good thing. In this context, the only
relevance of approximating JI is consonance, and only the simplest JI ratios
are consonant. More complex ratios, though they may be built out of simple
ratios, cannot be heard as such without the simple ratios actually present
in the music, and in the ET context they won't be. So right on!

I wrote,

>My work on harmonic entropy has allowed me to make these judgments more
precise and determine, under a given assumption >on the ear's tuning
resolution (I usually use 1% based on various research), the probability of
associating a particular ratio with a >given interval.

Shame on me! I've been seduced by Joe Monzo's incorrect definition of my own
concept in his dictionary! In fact, the probabilities come from a simple
model, essentially proposed by Van Eck. What harmonic entropy does is to
take those probabilities and compute a dissonance measure from them.