Re: [tuning] Re:JI and the listening composer - reply to Paul

Paul-Gene,

I'm not sure I'm getting this... wouldn't 25/24 be the smallest JI
interval derivable from two 5-limit consonances? Also, this seems (at
least how it's described here) like too broad and imprecise a thing...
I mean there's good and there's bad clustered there and there also
*are* other noteworthy 5-limit temperaments in-between.

I'm sure there must be more to it--or perhaps its rule of thumb can be
shown to be a very common generalization--for it to be as interesting
as you guys seem to be saying it is... how does this carry over to the
7-limit or the 11-limit, etc?

take care,

--Dan Stearns

----- Original Message -----
From: "emotionaljourney22" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Friday, June 14, 2002 2:12 PM
Subject: [tuning] Re:JI and the listening composer - reply to Paul

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> >
> > /tuning/topicId_37538.html#37706
> >
> > >> i'd be honored! personally, i like how suggests and explanation
> > the
> > > familiar bumps, like in 5-limit we have 19-22, 31-34, 41-43, 53-
> 55,
> > > pairs of good ets that cluster together in this regular pattern
> > with
> > > nothing noteworthy in-between. the 15:16 ratio is responsible
for
> > > this. the relative "power" of various superparticulars still
> seems
> > a
> > > bit of a mystery, though if graham is right, it comes down to
> > nothing more than a simple unweighted complexity metric . . .
> >
> >
> > ***I'll bet that was something discussed on *Tuning Math* since I
> > don't remember it from here. Could I please have a couple
> > of "layman's" sentences elaborating on this, if it's possible??
>
> well, this surely is an oversimplification, but it's sorta like
this:
>
> 16:15 is the *smallest* ji interval derivable from two 5-limit
> consonances (4:3 / 5:4 = 16:15).
>
> if a tuning has a very good approximation to the 5-limit
consonances,
> it will also have a pretty good approximation to 16:15.
>
> for an et to have a pretty good approximation to 16:15, the 16:15
> will have to be close to an integer number of steps in the ET.
>
> 16:15 fits in an octave 10.74 times.
>
> so you'd expect the good 5-limit ets to have numbers of notes close
> to multiples of 10.74 -- while ets *not* close to multiples of 10.74
> are unlikely to be good in the 5-limit.
>
> here are some multiples of 10.74:
>
> 10.74
> 21.48
> 32.22
> 42.96
> 53.70
>
> so you'd expect the good 5-limit ets to cluster around these values,
> and they do:
>
> 10, 12 are close to 10.74
> 19, 22 are close to 21.48
> 31, 34 are close to 32.22
> 41, 43 are close to 42.96
> 53, 55 are close to 53.70
>
>
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--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Paul-Gene,
>
> I'm not sure I'm getting this... wouldn't 25/24 be the smallest JI
> interval derivable from two 5-limit consonances?

just corrected that!

Also, this seems (at
> least how it's described here) like too broad and imprecise a
thing...
> I mean there's good and there's bad clustered there and there also
> *are* other noteworthy 5-limit temperaments in-between.
>
> I'm sure there must be more to it--or perhaps its rule of thumb can
be
> shown to be a very common generalization--for it to be as
interesting
> as you guys seem to be saying it is... how does this carry over to
the
> 7-limit or the 11-limit, etc?

see december 13 - december 15, on this list. the fourier transform of
the 7-limit goodness function of et cardinality has a *huge* peak at
a period of 1664, which is how many times a 2401:2400 fits into an
octave . . . 1664*62+1 = 103169 is what marc jones described as "used
as most convenient UHT to measure 7th limit intervals", and for good
reason!!