Re: MOS's, generators

Paul,

Some more replies, clarifications,

>>>Do we need to modify the definition of propriety? Would this modification
>>>have any bearing on your definition of the "best" generator?
>
>>I think you may be right. . . the first
>>question to ask would not be "is this scale proper" but "is this scale
>>proper in a system of n pitches per 8ve." The same goes for a set of
>>pitches described in a different mathematical system, such as the
>>pythagorean diatonic. This scale would be proper in a 12 tone system but
>>improper in a 17 note system. I'm taking for granted a method of
>>translation which I need to define more precisely. The criteria for a good
>>translation from, say, a set of heard pitches to a scale of n tones would
>>be that intervals between pitches which are heard as the same get the same
>>measurement on the scale.
>
>I don't think that criterion could help you decide whether n for a
>Pythagorean diatonic scale should be 12 or 17 (or 29, . . .)

That's not what the criterion is for. It's a criterion for determining
what are good fits, of which there are always more than one. The decision
of how to represent the Pythagorean diatonic could simply be, take the
lowest n which gives a good fit (since this is a translation between one
mathematical description and another, we have to give a tolerance in this
case. For translations from heard music to a representation, other factors
such as cultural indoctrination may also play a role, but I presume that we
will be able to specify by some other criteria which intervals must be the
same, and which different).

>> I would propose a different way of looking at this problem. I must
>say
>>that continued fractions don't strike me as the most useful way to
>>represent it.
>
>hm -- see below
>
>>They're useful for finding a noble number which the
>>successive generators approach, but for looking at proper scales which are
>>members of the sequence, I find the following representation simpler and
>>more descriptive. We're considering sequences of numbers which follow the
>>rule: the last member of the series at any point is the sum of the previous
>>number and some earlier number in the series.
>
>OK, why is that property important?

The weak criterion gives the set of all possible MOS's. The scales can be
represented by a sequence of larger and smaller steps, ie llsllls. If we
define l and s as any integers such that l > s, then the scale will fall
into some sequence satisfying the weak criteria. One value of this
representation is that it shows the russian-doll-like levels of structure
in the scales obtained. Its also more coherent with the approach to tuning
which I outlined in my last post. The continued fractions method supposes
a sequence of scales to be completed before any consideration of it, like a
Platonic mathematical entity. It starts with a generator with an exact
value and enumerates the sequence. But given my way of thinking about the
scales, it's meaningless to say without qualification that the generator
has an exact value. The pitch continuum is a distinct representational
frame: we can make translations from it into the discreet systems by the
best fit method, but they remain distinct. My representation of the
sequences is a process where a unique choice is made at each next step of
the sequence, and the last scale in the sequence demonstrates an embedded
pattern which is the sum total of each previous choice upstream. If the
choice is: how many places back do I go for a number to add to the last
one? then each choice (1, 2, 3, or . . . ) has a unique result in the ratio
of l:s in the next scale. The value of the generator is a result of the
process.

>>Then we consider sequences
>>where the latter number must be the number only two places previous in the
>>series.
>
>Have you been brainwashed by Yasser? :) :) :)

I suppose I should have addressed this when I started using Yasser's name.
Since he was, as far as I knew, the earliest person to propose the
sequence (of course, this is actually not entirely true), I put his name on
it. I certainly don't subscribe to his theories of harmony or evolution.
His ascription of ratios to the 13 limit excluding 3 to the degrees of
19-tET is absurd considering the fact that pure 3's come much closer to the
19-t degrees, aside from being more salient in general, than the 7's, 11's,
or 13's. There are many more absurdities there to talk about, but I don't
think it's worth going into.
Also, it's not out of the question that I may find it useful to diverge
from the strong criterion somewhat in taking the sequence beyond 31, so 2,
5, 7, 12, 19, 31, 43 is a possibility. You could also say 2, 3, 5, 7, . . .

>> The sequence with the least numbers per place satifying
>>the weaker criteria is the natural numbers. An interesting sequence
>>failing the stronger but satisfying the weaker criteria is the sequence of
>>MOS's of the pure 5th generator which Pierre mentioned, 2, 3, 5, 7, 12, 17,
>>29, 41 . . .
>
>The fact that it satisfies the weaker property is intimately intwined with
>the continued fractions I used to derive the sequence, in response to
>Pierre's challenge. Think about it! It should work for _any_ generator.

Of course. I didn't say that the continued fraction idea is useless.
That's one reason why I left the weaker criteria open, to show that all the
scales obtainable by the continued fraction method can also be described in
this fashion.

>Though, once again, I wonder why you're attracted to the strong criteria,
>I'm happy to note that all sequences that satisfy them will turn out to be
>the denominantors of the continued fraction convergents of noble numbers,
>and conversely.

Yes, and this fact gives us a useful method of finding the generator of
the scale at any step of the process. But here's where I cash in on the
observation above that each choice of a previous number of places to go
back to find the number to add to the last, given the weak criterion, gives
a unique l:s in the following scale: to explain the value of the strong
criteria. If the choice is always 1, then l:s is always 2:1. This gives
the scales of the highest stability relative to the same patterns where l:s
= 3:1 or higher, and I believe it also insures the propriety of the scales.
Also, if given a certain scalar pattern where the only obvious fact is
that there are larger and smaller steps, the simplest (system of least
resolution) way to represent that pattern is l = 2, s = 1. This minimizes
the information content of a pitch structure.

jason