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!!