Re: chain of minor thirds scale

>You've made a good case.

Well, I don't know about that.

>>You're using them, tho?
>
>I don't follow. How, or in what sense, am I using degree numbers?

By calling chords "subminor 6th", "diminished 7th", etc.

>Thanks for the Miller URL. Of course this talks about 7 +- 2 notes _total_
>in short term memory at any given time, not 7 +- 2 notes _per_octave_ in a
>scale.

Indeed. This was discussed on the list around Thanksgiving (can't remember
if we were on onelist yet). I don't think any conclusion was reached.

>But I guess it _is_ desirable for a whole octave to fit in STM.

I think there may be a feeling associated with keeping all the
octave-equivalent tones of a scale in STM, but I'm not sure if it's always
desirable, and I think the limit is probably more on the -2 side of 7 for
most people.

>>Rothenberg was able to explain the selection of 5 and 7 using only two
>>measures: stability and efficiency. Stability is essentially the
>>percentage of the scale's intervals which are "ambiguous", that is, which
>>appear in more than one interval class. For a definition of efficiency,
>>see TD #52 and 55.
>
>Thanks for this explanation and pointer too. But I'm not sure I understand
>them well enough to apply them to 4, 7, 8 and 11 of m3rds. Could you,
>please?

Stability is easy. Just count the intervals that are not involved in any
ambiguity and divide by the total number of intervals in the scale. So the
4-tone scale, like all strictly proper scales, has stability of 1. The
8-tone scale has stability of 0.35714, which is quite low. The 7 and 11
tone scales are improper, so stability is not defined.

Along with stability, I suggest two measures I call instability and
impropriety. The former is the portion of (octave-equivalent)
log-frequency space which is covered by intervals of the scale (using the
smallest and largest members of each scale degree as endpoints of a segment
on the line for that degree). The latter is the portion of the covered
area which is more than singly covered. When I've figured out how to get
Excel to draw these, I'll let you know.

The upshot here is that scales with low stability may tend to be used like
improper scales in music. So the Rothenberg portion of my argument
favoring the 8-tone scale over the 11-tone one is weak, pending further
listening and line graphs for each... the 11-tone version does have the
considerable advantage that the minor thirds are all confined to one
interval class...

Efficiency is the average number of tones you need to hear before you know
which key the scale is being played in, divided by the number of tones in
the scale. If n is the number of scale tones and S_i the length of a
key-minimal string, then efficiency is given by...

n!
(Sigma) S_i / n(n!)
i=1

...which is a bit of a pain. Rothenberg mentions in a footnote that
methods exist for calculating efficiency without having to find key-minimal
strings at all. Naturally, he refers to a paper that was never published.

>>What I like about Rothenberg's stuff is that he came up with it first,
>>actually for a speech recognition model, and then applied it to music and
>>got results that jived.
>
>Sounds great. I really want to learn this stuff, but I don't know when I'll
>make time to go and find the papers.

I should point out that to a large extent I am taking on faith his jiving
results. For convenience, he only considered scales as subsets of ETs.
The original papers give results for subsets of 12tET only, and say write
for further info. I met with Rothenberg in May. He said he had once made
a printout for every equivalence class in 31tET (!), and had made
comparisons with scales used around the world. He hasn't touched this
stuff in 30 years, however, and had trouble finding these supplementary
materials (and even the papers which I had brought).

>See what you think of it now.

Super! The 4-naturals notation looks to be my fav, except I would have
probably just used A-D. It was clever how you got the 8-naturals notation
to be alphabetical. Of course, you still have to live with accidentals
hopping scale degrees.

-C.