Re: Middle Paths, Gene, Graham, Paul

I thought what I was grinding away at had SOMEthing to do
with these threads, though I don't understand the terminology
being used (and it seems that complexity has multiple
definitions here!)

> Paul said
>
> > 2.1 1.2 1.1 1 2.1 1.2 1 1.1 2.1 1 2 1 1.1 2.1 1
>
> I don't think this was mentioned in Gene's very recent post on 31! It
> seems to be saying that a 9/31-oct. generator has an 11-limit
> complexity of only 15. Is this right, Gene?
>

> Graham replied
>
> Well, *I* make that a complexity of 19, so worse than meantone. Isn't it
> the usual neutral third scale?
>

Could Gene, Graham, Paul reach a concessus or straighten me out
one what complexity etc you are talking about?

The answer to Grahams point is, yes, this is an offspring of 45454545
but splatterring it out in this manner makes Jackys series available at
9 positions.

to which Gene replies

>
> This is 24+7, and it inherits the not-that-good 5 and worse 7 of the
> dreaded 24-et.

at which point I know we are talking in different languages. The
step sizes I am talking about are in 31et, which has excellent 5
and 7, and reasonable 3 and 11, and has nothing to do with 24et..

Meanwhile, just to show something practical about this particular
MOS in 31...

It is the sum of two meantone chromatic scales spaced apart
by 9 steps (approximate 11/9).

C Db D Eb E F F# G Ab A Bb B
0: 32323 23 32323 = 0 3 5 8 10 13 15 18 21 23 26 28
9: 32323 23 32323 = 9 12 14 17 19 22 24 27 30 1 4 6
-------------------------------------------------------
0 1 3 4 5 6 8 9 10 12 13 14 15 17 18 19 21 22 23 24 26 27 28 30

12111) (2112111) (2112111) (211) (21

It is probably a rule that any MOS can be expressed as a sum
of two other MOS. I'm not sure how useful this property is, but
I think it IS useful in this case since we've decomposed
something that may be somewhat unweildy to a sum of two well
understood structures (thinking musically, not mathematically
here).

Bob Valentine

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

> Could Gene, Graham, Paul reach a concessus or straighten me out
> one what complexity etc you are talking about?

We're all settled on Graham's definition, which defines the
n-complexity for an odd integer n to be the span of the generator
steps required to reach all odd numbers up to n. Hence if 3 requires
2, and 5 takes -7, (and 1, of course, 0) the 5-complexity would be 9,
since 1,3,5 fall into an interval of length 9 (containing therefore
10 integers.)

> > This is 24+7, and it inherits the not-that-good 5 and worse 7 of
the
> > dreaded 24-et.

> at which point I know we are talking in different languages. The
> step sizes I am talking about are in 31et, which has excellent 5
> and 7, and reasonable 3 and 11, and has nothing to do with 24et..

The 9/31 generator does have something to do with the 24-et--did you
read my reply to Paul? The badness of the 5 and 7 of the 24 and 7 ets
cancels to get the good *tuning* of the 31-et, but reinforces to get
the generators. The bad 5 and 7 translate into generators which
require a larger span--more complexity. Moreover the inconsistent
tendency--the fact that 24 has sharp 5s and flat 7s, and 7 is the
other way around--does not cancel, but rather reinforces, so that 7/5
is not well represented. What is out of tune in the other two ets
translates into what is not a characteristic interval of the
generator they define.

In-Reply-To: <9qm703+m4nd@eGroups.com>
Gene wrote:

> We're all settled on Graham's definition, which defines the
> n-complexity for an odd integer n to be the span of the generator
> steps required to reach all odd numbers up to n. Hence if 3 requires
> 2, and 5 takes -7, (and 1, of course, 0) the 5-complexity would be 9,
> since 1,3,5 fall into an interval of length 9 (containing therefore
> 10 integers.)

<cough>

Multiplied by the number of generators to the octave (or other equivalence
interval).

> The 9/31 generator does have something to do with the 24-et--did you
> read my reply to Paul? The badness of the 5 and 7 of the 24 and 7 ets
> cancels to get the good *tuning* of the 31-et, but reinforces to get
> the generators. The bad 5 and 7 translate into generators which
> require a larger span--more complexity. Moreover the inconsistent
> tendency--the fact that 24 has sharp 5s and flat 7s, and 7 is the
> other way around--does not cancel, but rather reinforces, so that 7/5
> is not well represented. What is out of tune in the other two ets
> translates into what is not a characteristic interval of the
> generator they define.

Why are you tarring 5 and 7 with the same feather? 7, 24 and 31 are all
easily consistent in the 11-limit-without-7. The neutral-third
temperament has a complexity of 8 in this limit, so two 11-limit-without-7
otonalities exist in the 10 note MOS. You also get one in Rast, expressed
in this temperament.

The more complex intervals are, the more they go out of tune as the
generator size drifts from the optimum. So that would explain what you
say about intervals "reinforcing". But intervals equally and oppositely
out of tune in different temperaments will also be in tune when you
combine them, which is useful to think about when you're looking for
temperaments to pair up.

I have the 7:1 mapping as -11 generator steps. Make it 20 generator
steps, and the 11-limit complexity is 20 instead of 10. As 7- and
24-equal aren't consistent in the 7-limit it isn't defined by h7+h24.

Graham

In-Reply-To: <200110180742.JAA87286@ius090.iil.intel.com>
Bob Valentine wrote:

> It is probably a rule that any MOS can be expressed as a sum
> of two other MOS. I'm not sure how useful this property is, but
> I think it IS useful in this case since we've decomposed
> something that may be somewhat unweildy to a sum of two well
> understood structures (thinking musically, not mathematically
> here).

Any MOS *with an even number of notes* within the period can be expressed
as two MOS scales with a generator twice as large. And you can generalize
it for any composite number, hence the three neutral third scales in
Blackjack, and the 6 meantone diatonics in Miracle72.

Graham

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
>
> > Could Gene, Graham, Paul reach a concessus or straighten me out
> > one what complexity etc you are talking about?
>
> We're all settled on Graham's definition, which defines the
> n-complexity for an odd integer n to be the span of the generator
> steps required to reach all odd numbers up to n. Hence if 3
requires
> 2, and 5 takes -7, (and 1, of course, 0) the 5-complexity would be
9,
> since 1,3,5 fall into an interval of length 9 (containing therefore
> 10 integers.)
>
> > > This is 24+7, and it inherits the not-that-good 5 and worse 7
of
> the
> > > dreaded 24-et.
>
> > at which point I know we are talking in different languages. The
> > step sizes I am talking about are in 31et, which has excellent 5
> > and 7, and reasonable 3 and 11, and has nothing to do with 24et..
>
> The 9/31 generator does have something to do with the 24-et--did
you
> read my reply to Paul? The badness of the 5 and 7 of the 24 and 7
ets
> cancels to get the good *tuning* of the 31-et, but reinforces to
get
> the generators. The bad 5 and 7 translate into generators which
> require a larger span--more complexity.

Oh -- so that's what you were talking about! Clearly you have a much
deeper insight into these things than I do . . . so how exactly do
the complexities "add" and why do they have anything to do with how
well the consonances are approximated in the parent tunings?

> Moreover the inconsistent
> tendency--the fact that 24 has sharp 5s and flat 7s, and 7 is the
> other way around--does not cancel, but rather reinforces, so that
7/5
> is not well represented.

Strange -- I would say that 7/5 is represented exceptionally well in
31. Do you mean "not well representes" in terms of quantity, rather
that quality?

> What is out of tune in the other two ets
> translates into what is not a characteristic interval of the
> generator they define.

I guess you do . . . fascinating!