612edo "semitones" (was: A common notation for JI and ETs)

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, February 25, 2002 7:59 PM
> Subject: [tuning] Re: A common notation for JI and ETs
>
>
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> /tuning/topicId_34739.html#34837
>
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > i think Gene meant, for example, 612edo, which gives 5-limit
> > > ratios with hardly any error and so is great as an integer
> > > measurement.
> >
> > It's actually a good 7, 9 and 11-limit system also.
>
>
> ***So... then we get 51 degrees for each "semi-tone"...?
>
> That may take a little "getting used to..." :)

well, Joe, actually, 612edo has a variety of different values
for the "semitone", depending on which semitone you mean!

612 / 12 = 51 , so yes, the 12edo semitone = exactly 51 degrees of 612edo.

but there are lots of other semitones which also work out to
be nearly exact integer values in 612edo. consult the lattices at
http://www.ixpres.com/interval/dict/eqtemp.htm

here are a few, which you can see on my lattices
(along with others); my boundaries for the "semitone"
are 50 - 150 cents, which is equivalent to 26 - 76 degrees
of 612edo.

(i've included only the ones closest to 1/1 in lattice space
for the 7- and 11-limit, and then only with +1 exponent for
those higher primes.)

[use "Expand Messages" to view this properly on the web.]

ratio vector ~cents 612edo degrees

5-limit

16875:16384 [ 3 4] 51.1 26
25:24 [-1 2] 70.7 36
256:243 [-5 0] 90.2 46
135:128 [ 3 1] 92.2 47
273375:262144 [ 7 3] 72.6 37
16:15 [-1 -1] 111.7 57
2187:2048 [ 7 0] 113.7 58
32768:30375 [-5 -3] 131.3 67
27:25 [ 3 -2] 133.2 68

7-limit

28:27 [-3 0 1] 63.0 32
21:20 [ 1 -1 1] 84.5 43

11-limit

33:32 [ 1 0 0 1] 53.3 27
275:256 [ 0 2 0 1] 123.9 63

notice the many pairs of intervals which are separated
by only one 612edo degree: that's because one degree of
612edo is nearly the same size as the skhisma, so these
pairs are separated by a skhisma and 612edo renders them
distinctly.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: jpehrson2 <jpehrson@r...>
> > To: <tuning@y...>
> > Sent: Monday, February 25, 2002 7:59 PM
> > Subject: [tuning] Re: A common notation for JI and ETs
> >
> >
> > --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> >
> > /tuning/topicId_34739.html#34837
> >
> > > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > i think Gene meant, for example, 612edo, which gives 5-limit
> > > > ratios with hardly any error and so is great as an integer
> > > > measurement.
> > >
> > > It's actually a good 7, 9 and 11-limit system also.
> >
> >
> > ***So... then we get 51 degrees for each "semi-tone"...?
> >
> > That may take a little "getting used to..." :)
>
>
>
> well, Joe, actually, 612edo has a variety of different values
> for the "semitone", depending on which semitone you mean!

monz, from the context of this, joseph obviously meant
the "standard", 100.0000-cent semitone.