Re: [tuning] re: "stability" and "efficiency"

Maybe Joe can add this to Carl's definition:

Rothenberg efficiency is defined as the average shortest
length of a random sequence of tones from a scale which is
needed to identify the key. If the scale size is N then
there are N! (N factorial) different sequences of all tones.
To calculate the efficiency, for all sequences the shortest
necessary length for key identification is summed and this
is divided by N! and again divided by N to get the relative
length, a number between 0 and 1. So efficiency measures how
many tones help you to identify the key. A low number means
there are many tones that don't help to do this. It may be
interpreted as a measure of the assymmetry of a scale with
respect to all rotations and translations of itself.
Rothenberg redundancy is defined as one minus the
efficiency. For scales with repeating blocks, like
Messiaen's modes, it is also defined with the difference
that the identified key will still be ambiguous.

I think also Winograd's "deep scale" property would make a
nice entry in the Monzo dictionary. Let me repeat the
definition that John Chalmers gave for it:

"Winograd called the scales that are generated by cycles of
any interval relatively prime to the octave (or any interval
of equivalence) "Deep Scales" if they have [C/2] or [C/2]+1
tones where C is the cardinality of the Interval of
Equivalence (tones per octave usually) and [C/2] is the
largest integer less than or equal to C/2. Such scales have
interval vectors that contain all the interval classes of
the chromatic set (C) with unique multiplicity (meaning that
an interval like as a major third occurs a unique number of
times compared to the number of times other intervals
appear). The interval vector is a vector whose elements are
the number of times each interval of the chromatic scale (C)
of which the scale is embedded or is a subset, appears.
Conventionally, the intervals are ordered from 1 degree to
[C/2]. For example, the interval vector of the diatonic
scale in 12-tet is [254361]. The interval vector of the
chromatic heptachord (0 1 2 3 4 5 6) is [654321] where the
generator, g, is 1 degree. Two other Deep Scales exist in
12-tET: the chromatic hexachord (0 1 2 3 4 5), whose interval
vector is [543210] and the hexatonic scale (0 2 4 5 7 9)
with interval vector [143250]. Winograd described these
relations originally in a hard to obtain term paper at MIT.
However, Carlton Gamer discussed them at length in "Some
Combinational Resources of Equal-Tempered Systems", _Journal
of Music Theory_ vol. 11 no. 1, 1967, pp. 32-59."

Manuel Op de Coul coul@ezh.nl

--- In tuning@egroups.com, <MANUEL.OP.DE.COUL@E...> wrote:
> http://www.egroups.com/message/tuning/12666
>
> [supplement to Carl's 'Rothenberg efficiency', and Chalmers's
> ['deep scale' definitions]

Thanks very much for the additional definitions, Manuel!
Also, thanks to Paul and John deL. for 'adaptive' definitions.

I'm currently having some phone problems, and can only access
the List at the public library, so it may be a day or two before
I can update both the Dictionary and the 'JI modulation' webpage.

-monz
http://www.ixpres.com/interval/monzo/homepage.html